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Emergence, Complexity and Computation ECC
Vincenzo Manca
Vincenzo Bonnici
Infogenomics
The Informational Analysis of Genomes

Emergence, Complexity and Computation
Volume 48
Series Editors
Ivan Zelinka, Technical University of Ostrava, Ostrava, Czech Republic
Andrew Adamatzky, University of the West of England, Bristol, UK
Guanrong Chen, City University of Hong Kong, Hong Kong, China
Editorial Board
Ajith Abraham, MirLabs, USA
Ana Lucia, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande
do Sul, Brazil
Juan C. Burguillo, University of Vigo, Spain
Sergej ˇCelikovský, Academy of Sciences of the Czech Republic, Czech Republic
Mohammed Chadli, University of Jules Verne, France
Emilio Corchado, University of Salamanca, Spain
Donald Davendra, Technical University of Ostrava, Czech Republic
Andrew Ilachinski, Center for Naval Analyses, USA
Jouni Lampinen, University of Vaasa, Finland
Martin Middendorf, University of Leipzig, Germany
Edward Ott, University of Maryland, USA
Linqiang Pan, Huazhong University of Science and Technology, Wuhan, China
Gheorghe P˘aun, Romanian Academy, Bucharest, Romania
Hendrik Richter, HTWK Leipzig University of Applied Sciences, Germany
Juan A. Rodriguez−Aguilar
, IIIA−CSIC, Spain
Otto Rössler, Institute of Physical and Theoretical Chemistry, Tübingen, Germany
Yaroslav D. Sergeyev, Dipartimento di Ingegneria Informatica, University of
Calabria, Rende, Italy
Vaclav Snasel, Technical University of Ostrava, Ostrava, Czech Republic
Ivo Vondrák, Technical University of Ostrava, Ostrava, Czech Republic
Hector Zenil, Karolinska Institute, Solna, Sweden

The Emergence, Complexity and Computation (ECC) series publishes new devel−
opments, advancements and selected topics in the ﬁelds of complexity, computa−
tion and emergence. The series focuses on all aspects of reality−based computation
approaches from an interdisciplinary point of view especially from applied sciences,
biology, physics, or chemistry. It presents new ideas and interdisciplinary insight on
the mutual intersection of subareas of computation, complexity and emergence and
its impact and limits to any computing based on physical limits (thermodynamic and
quantum limits, Bremermann’s limit, Seth Lloyd limits…) as well as algorithmic
limits (Gödel’s proof and its impact on calculation, algorithmic complexity, the
Chaitin’s Omega number and Kolmogorov complexity, non−traditional calculations
like Turing machine process and its consequences,…) and limitations arising in arti−
ﬁcial intelligence. The topics are (but not limited to) membrane computing, DNA
computing, immune computing, quantum computing, swarm computing, analogic
computing, chaos computing and computing on the edge of chaos, computational
aspects of dynamics of complex systems (systems with self−organization, multiagent
systems, cellular automata, artiﬁcial life,…), emergence of complex systems and its
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to discuss the above mentioned topics from an interdisciplinary point of view and
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notes, selected contributions from specialized conferences and workshops, special
contribution from international experts.
Indexed by zbMATH.

Vincenzo Manca <> · <> Vincenzo Bonnici
Infogenomics
The Informational Analysis of Genomes

Vincenzo Manca
Department of Computer Science
University of Verona
Pisa, Italy
Vincenzo Bonnici
Department of Mathematics, Physics and Computer Science University of Parma Parma, Italy
ISSN 2194−7287 ISSN 2194−7295 (electronic)
Emergence, Complexity and Computation
ISBN 978−3−031−44500−2 ISBN 978−3−031−44501−9 (eBook)
https://doi.org/10.1007/978−3−031−44501−9
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2023
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transmission or information storage and retrieval, electronic adaptation, computer software, or by similar
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Paper in this product is recyclable.

We dedicate this book to our nearest nodes in
the tree of life to which we belong: our
parents, our children, and grandchildren (of
the oldest author).

Preface
This book originates in the courses that authors took in the last ten years at the
universities of Verona and Parma, and before at the University of Udine and Pisa.
They have been on very different subjects centered on information, in many scien−
tiﬁc ﬁelds: computation, formal representation, unconventional computing, molec−
ular biology, artiﬁcial intelligence. The names of courses were: Information Theory,
Languages and Automata, Unconventional Computing, Discrete biological Models,
Programming Laboratory, Data Structures, Computational Biology. The book does
not collect all the topics of these courses, but surely inﬂuenced by them, it is focused
on genomes.
The conviction that we maturated during our teaching and related research, during
last years, was the centrality of information as the common factor of all subjects of our interests. Information is the most recent scientiﬁc concept. It emerged in the last century, by revolutionizing the whole science. The reason of this is evident.
Namely, all scientiﬁc theories develop by collecting data from observed phenomena,
and these data support theories which explain phenomena and provide new scientiﬁc
knowledge.
Information is inside data, but it does not coincide with them. It assumes and
requires data, in order to be represented, but is independent from any speciﬁc data representation, because it can be always equivalently recovered by encoding data
into other data.
This is the reason of its deep almost evanescent nature, but also of its ﬂuidity, ﬂexi−
bility, adaptability, and universality. Information can always be linearized into strings. Genomes, which are the texts of life, are strings. But these linear forms of repre−
sentation are managed, elaborated, transformed, and become meaningful, through
networks, made of elements connected by relations, at different levels of complexity.
Our brains are a particular case of such networks which develops through sensory,
linguistic, social, and cultural information. In the dynamics of the communication,
the ethereal essence of information ﬂows and determines the various kinds of reality
of our knowledge universe.
vii

viii<> Preface
This book tells small fragments of this big story, that we collected and partially
understood during our scientiﬁc adventure.
Pisa, Italy
Parma, Italy
Vincenzo Manca
Vincenzo Bonnici
Acknowledgments We express our gratitude to the students of the universities of Verona, Pisa,
and Parma, which in the last ten years followed our courses in the curricula of Computer Science,
Biological Sciences, Computational Biology, and Mathematics, at different levels. Their comments
and suggestions improved the quality of the notes from which this book originates.We thank also
our colleagues: Giuditta Franco, Giuseppe Scollo, Luca Marchetti, Alberto Castellini, Roberto
Pagliarini, Rosario Lombardo, of the University of Verona, who collaborated with us in writing
many papers in the research area of the book.

Contents
1 The Infogenomics Perspective...................................1
References.....................................................4
2 Basic Concepts.................................................7
2.1Sets and Relations.........................................7
2.2Strings and Rewriting......................................12
2.3Variables and Distributions.................................16
2.4Events and Probability.....................................16
References.....................................................22
3 Information Theory............................................23
3.1From Physical to Informational Entropy......................24
3.2Entropy and Computation..................................27
3.3Entropic Concepts.........................................31
3.4Codes...................................................38
3.5Huffman Encoding........................................43
3.6First Shannon Theorem....................................47
3.7Typical Sequences.........................................50
3.7.1 AEP (Asymptotic Equipartition Property) Theorem......50
3.8Second Shannon Theorem..................................52
3.9Signals and Continuous Distributions.........................55
3.9.1 Fourier Series......................................57
3.10 Fourier Transform.........................................61
3.11 Sampling Theorem........................................62
3.12 Third Shannon Theorem....................................64
References.....................................................65
4 Informational Genomics........................................67
4.1DNA Structure............................................68
4.2Genome Texts............................................74
4.3Genome Dictionaries......................................77
4.4Genome Informational Indexes..............................79
ix

x<> Contents
4.5Genome Information Sources...............................82
4.6Genome Spectra...........................................86
4.7Elongation and Segmentation...............................91
4.8Genome Informational Laws................................97
4.9Genome Complexity.......................................98
4.10 Genome Divergences and Similarities........................100
4.11 Lexicographic Ordering....................................105
4.12 Sufﬁx Arrays.............................................110
References.....................................................111
5 Information and Randomness...................................113
5.1Topics in Probability Theory................................113
5.2Informational Randomness.................................133
5.3Information in Physics.....................................140
5.4The Informational Nature of Quantum Mechanics..............151
References.....................................................156
6 Life Intelligence................................................159
6.1Genetic Algorithms........................................162
6.2Swarm Intelligence........................................164
6.3Artiﬁcial Neural Networks..................................165
6.4Artiﬁcial Versus Human Intelligence.........................175
References.....................................................186
7 Introduction to Python..........................................189
7.1The Python Language......................................189
7.2The Python Environment...................................190
7.3Operators................................................194
7.4Statements...............................................195
7.5Functions................................................197
7.6Collections...............................................200
7.7Sorting...................................................209
7.8Classes and Objects........................................210
7.9Methods.................................................211
7.10 Some Notes on Efﬁciency..................................215
7.11 Iterators..................................................216
7.12 Itertools..................................................219
8 Laboratory in Python...........................................223
8.1Extraction of Symbols.....................................223
8.2Extraction of Words.......................................225
8.3Word Multiplicity.........................................226
8.4Counting Words...........................................228
8.5Searching for Nullomers...................................230
8.6Dictionary Coverage.......................................233
8.7Reading FASTA Files......................................235
8.8Informational Indexes......................................236

Contents<> xi
8.9Genomic Distributions.....................................239
8.10 Genomic Data Structures...................................249
8.11 Recurrence Patterns........................................270
8.12 Generation of Random Genomes............................278
References........................................................283
Index.............................................................285

Acronyms
2LE Binary Logarithmic Entropy
AC Anti−entropic Component
AEP Asymptotic Equipartition Property
AI Artiﬁcial Intelligence
AMD Average Multiplicity Distribution
ANN Artiﬁcial Neural Network
BB Bio Bit
CSO Constraint Optimization Problem
EC Entropic Component
ECP Entropic Circular Principle
EED Empirical Entropy Distribution
EP Equipartition Property
ESA Enhanced Sufﬁx Arrays
ghl Greatest Minimal Hapax Length
GPT Generative Pre−trained Transformer
LCP Longest Common Preﬁx
LE Logarithmic Entropy
LG Logarithmic Length
LX Lexical Index
mcl Maximal Complete Length
mﬂ Minimum Forbidden Length
mhl Minimum Hapax Length
mrl Maximum Repeat Length
nhl No Hapax Length
nrl No Repeat Length
PSO Particle Swarm Optimization
RDD Recurrence Distance Distribution
WCMD Word Co−Multiplicity Distribution
WLD Word Length Distribution
WMD Word Multiplicity Distribution
xiii

Chapter 1
The Infogenomics Perspective
Prologue
This book presents a conceptual and methodological basis for the mathematical and
computational analysis of genomes, as developed in publications [ 1 – 13]. However,
the presentation of the following chapters is, in many aspects, independent from their
original motivation and could be of wider interest.
Genomes are containers of biological information, which direct the cell func-
tions and the evolution of organisms. Combinatorial, probabilistic, and informational
aspects are fundamental ingredients of any mathematical investigation of genomes
aimed at providing mathematical principles for extracting the information that they
contain.
Basic mathematical notions are assumed, which are recalled in the next chapter.
Moreover, basic concepts of Chemistry, Biology, Genomics [ 14], Calculus and Prob-
ability theory [ 15 – 17] are useful for a deeper comprehension of several discussions
and for a better feeling of the motivations that underlie many subjects considered in
the book.
A key concept of this book is that of distribution, a very general notion occurring
in many forms. The most elementary notion of distribution consists of a set and a
“multiplicity” function that associates a number of occurrences to each element of
the set. Probability density functions or distributions in the more general perspective
of measure theory generalize this idea. Distributions as generalized functions are
central in the theory of partial differential equations. The intuition common to any
notion of distribution is that of a “quantity” spread over a “space”, or parts, of a whole
quantity, associated to locations. A probability distribution is a way of distributing
possibilities to events. Probability and Information are for many aspects two faces
of the same coin, and the notion of “Information Source”, which is the starting point
of Shannon’s Information Theory, is essentially a probability distribution on a set of
data.
The topics presented in the book include research themes developed by authors
in the last ten years, and in many aspects, the book continues a preceding volume
published by Vincenzo Manca, Infobiotics: Information in biotic systems, Springer,
2013 [ 4]. We could shortly say that, abstractly and concisely, the book [ 4] investigates
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
V. Manca and V. Bonnici, Infogenomics, Emergence, Complexity and Computation 48,
https://doi.org/10.1007/978-3-031-44501-9_1
1

2 1 The Infogenomics Perspective
the relationship between rewriting and metabolism (metabolic P systems are the
formal counterpart of metabolic systems), while the present book investigates the
relationship between distributions and genomes.
The main inspiring idea of the present book is an informational perspective to
Genomics. Information, is the most recent, among the fundamental mathematical
and physical concepts developed in the last two centuries [ 18, 19]. It has revolu-
tionized the whole science, and continues, in this direction, to dominate the trends
of contemporary science. In fact, any discipline collects data from observations, by
providing theories able to explain, predict, and dominate natural phenomena. But
data are containers of information, whence information is essential in any scientiﬁc
elaboration.
The informational viewpoint in science becomes even more evident for the sci-
ences of life, which after molecular biology, have revealed as living organisms elab-
orate and transmit information, organized in molecular structures. Reproduction is
the way by means of which information is transmitted along generations, but, in the
same moment that biological information is transmitted, random variations alter a
small percentage of the message passed to the reproduced organism.
Any reproductive system is reﬂexive, in the sense that it includes a proper part
containing the information for reconstructing the whole organism. Thus, reﬂexivity is
a purely informational concept, and DNA molecules are the basilar forms of reﬂexive
structures, where reﬂexivity is realized by pairing two strands of biopolymers that
can be recovered, by a template-driven copying process, from only one of them.
Genomes evolved from very short primordial biopolymers, proto-genomes, that
assumed the function of biological memories. Enzymes, which direct, as catalysts,
metabolic reactions, are the ﬁrst molecular machines, but as any physical com-
plex structure, they are subjected to destructive forces acting over time. The only
way of keeping their speciﬁc abilities is the possibility of maintaining traces of
them. Thus, proto-genomes probably originate as enzymatic memories, sequences
including “copies” of the enzymes of proto-cells (membrane hosting substances and
reactions among them). These memories, passing along generations, allowed life
to develop and evolve, without losing the biological information cumulated along
generations.
Genomes are “long” strings, or texts, of millions or billions of letters. These texts
describe forms of life and even the history of life that produced them. Therefore,
genomes are biologically meaningful texts. This means that these texts postulate
languages. The main challenge for the biology of the next centuries is just discover-
ing these languages. Speciﬁc steps in this direction are the following: (1) to identify
the minimal units conveying information and their categories; (2) to discover the
syntax expressing the correct mechanisms of aggregation of genomic units, at their
different levels of articulation; (3) to understand the ways and the contexts activating
statements that direct the (molecular) agents performing speciﬁc cellular tasks; (4)
to determine the variations admitted in the class of individual genomes of a given
species; (5) to reconstruct the relationships between genome structures and the cor-
responding languages that these structures realize in the communication processes
inside cells.

1 The Infogenomics Perspective 3
Genomes are texts, but in a very speciﬁc sense, which is hardly comparable with
the usual texts telling stories or describing phenomena. Maybe, a close comparison
could be done with a legal code, prescribing rules of behaviour, with a manual of the
use of a machine, or with an operative system of a computer. However, this special
kind of manual/program is a sort of “living manual”, cabled in the machine, as the
interpreter of a programming language, but able to generate new machines. This fact
is directly related to the cruciality of reproduction. Genomes generate statements
assimilable to linguistic forms of usual languages, which instruct cell agents and, in
some particular contexts, can change just parts of the same genomes directing the
play, but somewhat paradoxically, they maintain a memory of their past history, but
also of the mechanisms allowing their future transformations.
In fact, genomes structures and related languages provide an intrinsic instabil-
ity promoting a small rate of change pushing toward new structures. This means that there are general genome built-in rules of internal variation and reorganization.
Genealogies of life are ﬁrstly genealogies of genomes, very similar to those of nat-
ural languages. In this way, from a biological species new species derive, through
passages called <> speciations<>. A new species arises when individual variations, in
restricted classes of individuals, cumulate in such a way that some internal criteria of
coherence require a new organization incompatible with the genome of the ancestor
organism.
These dynamics require two kinds of interactions, one between species and indi-
viduals, and another between syntax and semantics of the genome languages. The genome of a species is an abstract notion, because, in the concrete biological reality, only individual genomes exist, and the genome of a species is the set of genomes of
all individuals of that species. If some speciﬁc attributes have a range of variability, in
the genome of the species these attributes become variables. Such a kind of abstract
genome is a pattern of individual genomes. Fisher’s theorem on natural selection (due
to Ronald Fisher, 1890–1962) establishes that the time variation rate of genomes, of
a given species, is proportional to the genome variability among the individuals of
that species. In other words, the more genomes are variegate in the individuals living
at some time, the more, on average, they change along generations.
Variations in individual genomes promote new interactions between syntax and
semantics of genome languages, where semantics “follows” syntax. In other words, syntax admits some statements that even if well-formed can be meaningless. How-
ever, under the pressure of some circumstances, certain statements, casually and
temporarily, assume new meanings that in cell interactions “work”, becoming seeds
of new semantic rules. If these rules are reinforced by other situations of success in
cellular communication, they eventually stabilize. Speciation occurs when the num-
ber of genome variations overcomes a given threshold so that the language underlying
the genomes results differently with respect to those of ancestor genomes. At that
moment, the resulting organisms represent a new branch in the tree of life.
This “linguistic” scenario is only evocative of possible investigations, however,
it is a good argument for explaining why considering genomes as “texts of life” is fascinating and could be one of the most promising approaches to the comprehension
of the evolutionary mechanisms. In this regard, a clue about <> life intelligence <> resides

4 1 The Infogenomics Perspective
in the relationship between chance and purpose. Randomness provides a powerful
generator of possibilities, but the navigation in the space of possibilities selects tra-
jectories that result in more convenience to realize some ﬁnalities. Probably, this form
of intelligence, is spread evenly in the primordial genomes (as rules of syntactical
coherence?). This possibility could explain the efﬁciency of evolution in ﬁnding the
best solutions for life development. This theme is present since Wiener’s pioneering
research in Cybernetics (“Purpose, Finality and Teleology” is the title of Wiener’s
paper in 1943, with Arturo Rosenblueth and Julian Bigelow, anticipating his famous
book published in 1948). Random genomes, which will be considered in the chapter
“Information and Randomness” of this book, show a purely informational property
of random texts.
This short introduction wants to emphasize the role that information plays in
biological systems. The awareness of this fact persuaded us that a deep understanding
of life’s fundamental mechanisms has to be based on the informational analysis of
biological processes. In this spirit, the following chapters outline general principles
and applications of information theory to genomes.
Finally, a general remark on information concerns its relationship with knowl-
edge. It is important to distinguish between the two notions. Knowledge is surely
based on information, but a collection of data is something completely different from
knowledge. Knowledge appears when data are interpreted within theories where they
ﬁt, by revealing an internal logic connecting them. But theories do not emerge auto-
matically from data, conversely, data are explained when they are incorporated within
theories, which remain scientiﬁc creations of human (or artiﬁcial?) intelligence.
References
1. Manca, V.: On the logic and geometry of bilinear forms. Fundamenta Informaticae 64, 261–
273 (2005)
2. Manca, V., Franco, G.: Computing by polymerase chain reaction. Math. Biosci. 211, 282–298
(2008)
3. Castellini, A., Franco, G., Manca, V.: A dictionary based informational genome analysis.
BMC Genomics 13, 485 (2012)
4. Manca, V.: Infobiotics: Information in Biotic Systems. Springer, Berlin (2013)
5. Manca, V.: Infogenomics: Genomes as Information Sources. Chapter 21, pp. 317–324. Else-
vier, Morgan Kauffman (2016)
6. Bonnici, V., Manca, V.: Infogenomics tools: a computational suite for informational analysis
of genomes. J. Bioinform. Proteomics Rev. 1, 8–14 (2015)
7. Bonnici, V., Manca, V.: Recurrence distance distributions in computational genomics. Am. J.
Bioinform. Comput. Biol. 3, 5–23 (2015)
8. Manca, V.: Information theory in genome analysis. In: Membrane Computing, LNCS, vol.
9504, pp. 3–18. Springer, Berlin (2016)
9. Bonnici, V.: Informational and Relational Analysis of Biological Data. Ph.D. Thesis, Dipar-
timento di Informatica Università di Verona (2016)
10. Bonnici, V., Manca, V.: Informational laws of genome structures. Sci. Rep. 6, 28840 (2016).
http://www.nature.com/articles/srep28840. Updated in February 2023
11. Manca, V.: The principles of informational genomics. Theor. Comput. Sci. (2017)

References 5
12. Manca, V., Scollo, G.: Explaining DNA structure. Theor. Comput. Sci. <> 894, 152–171 (2021)
13. Bonnici, V., Franco, G., Manca, V.: Spectrality in genome informational analysis. Theor.
Comput. Sci. (2020)
14. Lynch, M.: The Origin of Genome Architecture. Sinauer Associate. Inc., Publisher (2007)
15. Aczel, A.D.: Chance. Thunder’s Mouth Press, New York (2004)
16. Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)
17. Schervish, M.J.: Theory of Statistics. Springer, New-York (1995)
18. Brillouin, L.: Scienze and Information Theory. Academic, New York (1956)
19. Brillouin, L.: The negentropy principle of information. J. Appl. Phys. <> 24, 1152–1163 (1953)

Chapter 2
Basic Concepts
2.1 Sets and Relations
A set is a collection of elements considered as a whole entity characterized only and
completely by the elements belonging to it. Braces are used for enclosing a ﬁnite list
of elements of a set, or for enclosing a variable. xand (after a bar. |) a property. P<>(<>x<>)
true when. xassumes values that are elements of the set:.{<>x<> |<> P<>(<>x<>)} .
Two basic relations. ∈and.⊆denote when an element belongs to a set, and when
a set is included in another set (all elements of the ﬁrst set belong also to the second
set). A special set is the empty set denoted by. ∅that does not contain any element.
On sets are deﬁned the usual boolean operations of union . ∪, intersection . ∩and
difference–. The powerset.P(<>A<>)is the set of all subsets of. A.
The pair set.{<>a<>,<> b<>}has only. aand. bas elements.
The ordered pair.(<>a<>,<> b<>)is a set with two elements, but where an order is identiﬁed.
A classical way to express ordered pair is the following deﬁnition due to Kuratowski:
. (<>a<>,<> b<>)<> ={<>a<>,<> {<>a<>,<> b<>}}
where the ﬁrst element. ais the element that is a member of the set and of the innermost
set included in it, whereas the second element. bbelongs only to the innermost set.
The cartesian product.A<> ×<> Bof two sets is the set of ordered pairs where the ﬁrst
element is in. Aand the second is in. B.
The notion of ordered pair can be iteratively applied for deﬁning.n-uples for any
. nin the set. Nof natural numbers. In fact, an.n-uple is deﬁned in terms of pairs:
. (<>a1,<> a2,<> a3,...,<>an)<> =<> (<>a1,(<>a2,(<>a3,...a n))).
Logical symbols .∀,<> ∃,<> ∨,<> ∧,<> ¬,<> → abbreviate <> for all, there exists, or, and, not,
if-then<>, respectively.
A.k-ary relation. Rover a set. Ais a set of.k-uples of elements of. A.. R<>(<>a1,<> a2,...,<>ak)
is an abbreviation for.(<>a1,<> a2,...,<>ak)<> ∈<> R.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
V. Manca and V. Bonnici, <> Infogenomics<>, Emergence, Complexity and Computation 48,
https://doi.org/10.1007/978-3-031-44501-9_2
7

8 2 Basic Concepts
Binary relations are of particular importance and among them equivalence and
ordering relations, with all the concepts related to them. Often.R<>(<>a<>,<> b<>)is abbreviated
by.aRb.
A binary relation over a set .Ais an equivalence relation over .Aif the following
conditions hold:
.aRa<> ∀<> a<> ∈<> A (Refexivity)
.aRb<> ⇒<> bRa<> ∀<> a<>,<> b<> ∈<> A (Symmetry)
.aRb<>∧<> bRc<> ⇒<> aRc<> ∀<> a<>,<> b<>,<> c<> ∈<> A (Transitivity)
Given an equivalence class over a set. Aand an element. aof. Athe set. [<>a<>]R={<>x<> ∈
A<> |<> aRx<>} is the equivalence class of. a.The set.{[<>x<>]R|<> x<> ∈<> A<>} is the quotient set of
. Awith respect to . R, denoted by .A<>/R. This set is a partition of . A, that is, the union
of all its sets coincides with .A(it is a covering of . A) and these sets are non-empty
and disjoint (with no common element).
A binary relation over a set .Ais an ordering relation over .Aif the following
conditions hold:
.aRa<> ∀<> a<> ∈<> A (Reﬂexivity)
.aRb<>∧<> bRa<> ⇒<> a<> =<> b<> ∀<> a<>,<> b<> ∈<> A (Antisymmetry)
.aRb<>∧<> bRc<> ⇒<> aRc<> ∀<> a<>,<> b<>,<> c<> ∈<> A (Transitivity)
An ordering relation is partial if two elements exist such that both.aRband. bRa
do not hold. In this case,.a<>,<> bare incomparable with respect to. R. An order is linear
or total if for all two elements .a<>,<> b<> ∈<> Aat least one of the two cases .aRband . bRa
holds. Given a subset. Bof. A, the element.m<> ∈<> Bis the minimum of. Bfor. Rif. mRb
for all.b<> ∈<> B, while. mis minimal in. Bif no.b<> ∈<> Bexist such that.bRm. Analogously
.M<> ∈<> Bis the maximum of. Bfor. Rif.bRMfor all.b<> ∈<> B, and.Mis maximal in. Bif
no.b<> ∈<> Bexists such that.MRb.
Given a subset. Bof. Athe element.m<> ∈<> Ais a lower bound of. B, with respect to. R,
if.mRbfor all.b<> ∈<> B. Analogously.M<> ∈<> Bis an upper bound of. Bwith respect to. R
if.bRMfor all.b<> ∈<> B. In these cases,. mis the greatest lower bound of. Bwith respect
to .Rif .mis the maximum in the set of the lower bounds of . B, and analogously . M
is the lowest upper bound of. B, with respect to. R,if.Mis the minimum in the set of
the upper bounds of. B.
A .k-operation .ωover a set .Ais a .(<>k<> +<> 1<>)-ary relation over .Awhere the last
argument depends completely on the others. In other words, if two .(<>k<> +<> 1<>)-uples
of the relation coincide on the ﬁrst . kcomponents, then they coincide on the last
argument too. The last argument is also called the result of the operation . ωon the
ﬁrst. karguments.a1,<> a2,...,<>akand is denoted by.ω(<>a1,<> a2,...,<>ak).
A function of domain. Aand codomain. B
.f<> :<> A<> →<> B

2.1 Sets and Relations 9
(from .Ato . B) is identiﬁed by the two sets .A<>,<> Band by an operation .fsuch that
when. fis applied to an element. aof. A, then the result.f<> (<>a<>)is an element of. B,also
called the image of . aaccording to . fand if .f<> (<>a<>)<> =<> b, element . ais also called the
counter-image or inverse image of. b. A function is injective when distinct elements
of the domain have always distinct images in the codomain. A function is surjective
when all the elements of .Bare images of elements of . A, A function is 1-to-1 or
bijective when it is injective and surjective. Given a subset.C<> ⊆<> Bthe set of inverse
images.{<>x<> ∈<> A<> |<> f<> (<>x<>)<> ∈<> C<>} is denoted by.f
−<>1
(<>C<>).
Set-theoretic concepts and notation are the basic language of mathematics, devel-
oped in the last two centuries [ 1, 2]. Within the set theory, Cantor developed a rigor-
ous mathematical analysis of inﬁnity, by showing the rich articulation of this concept when it is mathematically formalized (transﬁnite, ordinal and cardinal numbers are sets). Set theory is also the basis of general topology, where general notions of space
can be developed. Nowadays, almost all mathematical theories can be expressed in
the set-theoretic language (see [
3, 4] for introductory texts to set theory). For the
needs of our discourse, the basilar concepts given here are enough. The foundational
power of sets will not appear explicitly in the chapters of this book. However, it is
important to realize that the expressiveness and universality of set-theoretic language
is mainly due to the conceptual strength of the set theory, which in its advanced topics
remains one of the most complex and fascinating mathematical theories.
A ﬁnite sequence of length .nover a set .Ais a function from (positions)
.{<>1<>,<> 2<>,...,<>n<>} to. A.
A ﬁnite multiset of size. nover a set. Acan be considered as a function from. Ato
natural numbers assigning a multiplicity to any element of the set . A, such that the
sum of all multiplicities is equal to. n.
An inﬁnite sequence, denoted by .(<>an|<> n<> ∈<> N<>) has the set of natural numbers as
a domain. The same notation with a generic set. Iof indexes denotes a family over a
set. A(indexed by. I). An alphabet. Ais a ﬁnite set of elements, called symbols. Finite
sequences of symbols are also called strings.
A variable is an element. xwhich is associated with a set. A, called the range of. x,
where the variable takes values in correspondence to some contexts of evaluation. The notion of variable is strictly related to that of function, but in a variable is stressed its range rather than the correspondence between contexts of evaluation (corresponding
to the domain of a function) and the values assumed in the range (corresponding to
the codomain of a function). In fact, for a variable, often, the mechanism determining
the values is unknown or is not relevant.
If a variable . xtakes values in correspondence to the values of another variable,
for example, a variable. tof time, then the value assumed by. xare denoted by.x<>(<>t<>),
and the correspondence,
. t<> |→<> x<>(<>t<>)
determines a function from the range of. tto the range of. x.
When we apply operations to elements, in a composite way, for example:
.ω(<>x<>,η(<>y<>)))

10 2 Basic Concepts
where. ωis a binary operation and. ηis a unary operation, then the expression above
denotes a value in correspondence to the values of variables .x<>,<> y. Parentheses are
essential for denoting the order of applications of operations. In the case of the
expression above the innermost parentheses are related to the application of . ηand
after determining the value of .η(<>y<>))the operation .ωis applied to its arguments.
In this case, a function can be deﬁned as having as domain the cartesian product of the ranges of
. xand . yand, as codomain, the values assumed by .ω(<>x<>,η(<>y<>))) in
correspondence to the values of. xand. y:
. (<>x<>,<> y<>)<> |→<> ω(<>x<>,η(<>y<>))).
The deep understanding of the relationship among arithmetic, logic, and com-
putability is one of the most exciting successes of mathematical logic in the 20th
century, explaining the pervasive role of symbolic sequences in computation pro-
cesses [ 5, 6]. In a sense, molecular biology discovered the same kind of pervasive-
ness for biopolymers, which are representations of symbolic sequences at a molecular
level.
The notion of expression can be mathematically generalized by the notion of
a rooted tree, which expresses the abstract notion underlying a genealogical tree.
A rooted tree is essentially obtained by starting from an element, called root, by
applying to it a generation operation providing a set of nodes, that are the sons of the
root, and then by iteratively applying to generated nodes the generation operation.
When no generation operation is applied to a node, then it is a leaf. An element that
is not a leaf is an internal node. A sequence of elements where any element (apart
from the ﬁrst) is the son of the previous one provides a branch of the tree, and a tree
is inﬁnite if it has an inﬁnite branch.
Trees can be further generalized by graphs. A graph is given by a set. Aof elements,
called nodes, and a set .Eof edges such that each edge is associated with a source
node and a target node. A path in a graph is a sequence of nodes where for each node
. aof the sequence, apart from the last element of the sequence, exists another node
. bsuch that . aand . bare the source and the target of an edge of the graph. A path is
inﬁnite if no last element is present in the sequence, and it is a cycle if the last node of the path is equal to its ﬁrst node. A graph is connected if, for any two nodes of
the graph, there exists a path starting from one node and ending in the other one. A
graph that is connected and without cycles is called a non-rooted tree.
Basic combinatorial schemata
Combinatorics deals with problems of counting ﬁnite structures of interest in many
mathematical contexts [ 7]. These structures are usually deﬁnable in terms of sets,
sequences, functions, and distributions over ﬁnite sets. Counting them is often not
only mathematically interesting in itself, but very important from the point of view
of a lot of applications. Combinatorics was the starting point of probability because

2.1 Sets and Relations 11
some events correspond to the occurrence of structures of a given subclass within a
wider class of possible structures. In many cases, a ﬁnite structure of interest is given
by a <> distribution <> of objects into cells.
The number of functions from. nobjects to. mcells is.n
n
(any object can be allocated
in one of the cells), whereas the number of bijective functions from . nobjects to . n,
called also.n-<>permutations<>, is obtained by allocating the ﬁrst object in. nways, the
second one in.n<> −<> 1ways (because cannot be allocated in the ﬁrst cell), and so forth,
with the last object allocated in only one possible way (in the last remained cell). Therefore:
. |<>n<> ↔<> n<>|=<>n<>!<>.
where.n<>!is the factorial of. n, that is, the product.1<> ×<> 2<> ×···×<>(<>n<> −<> 1<>)<> ×<> n .
Reasoning as in the previous cases the number of injective functions . |<>n⊂→<> m<>|
from. nto.mobjects (.n<> ≥<> m) is given by:
. |<>n⊂→<> m<>|=<>n<>! m
where.n<>!mis the <> falling factorial <> of .mdecreasing factors starting from. n:
. n<>!m=<> n<>(<>n<> −<> 1<>)(n<> −<> 2<>)...(<>n<> −<> m<> +<> 1<>)<> =
m
∏
k<>=<>1
(<>n<> −<> k<> +<> 1<>)
Counting the number of .m-subsets of . nelements can be obtained by counting
all the injective functions.n<> →⊃mand then ignoring the distinguishability of cells,
by dividing by .m<>!, which corresponds to all the possible ways of ordering cells. In
conclusion, the number.|<>m<> ⊆<> n<>|of.m-sets is given by:
. |<>m<> ⊆<> n<>|=<>n<>! m/<>m<>!<>.
The.m-sets over. nobjects are called <> combinations <> of . mover. n, and their numbers
.n<>!m/<>m<>!are called <> binomial coefﬁcients<>, and denoted by:
.
(
n
m
)
=<> n<>!
m/<>m<>!
also given by the following formula:
.
(
n
m
)
=
n<>!
m<>!<>(<>n<> −<> m<>)<>!
.
The number of <> partitions of . nundistinguishable objects <> in . kdistinct classes,
possibly empty, can be obtained by considering boolean sequences of . nzeros and
.k<> −<> 1ones. A boolean sequence of .n<> +<> k<> −<> 1 positions is completely determined
when we chose the. npositions where put zeros, therefore:

12 2 Basic Concepts
.
(
n<> +<> k<> −<> 1
n
)
=<> (<>n<> +<> k<> −<> 1<>)
!<>n/<>n<>!
If we use the <> rising factorial <> notation .k<>!
n
(the product of. nincreasing factors starting
from. k) it is easy to realize that:
.
(
n<> +<> k<> −<> 1
n
)
=<> k<>!
n
/<>n<>!
If we choose. kobjects, over. ndistinct objects, with the possibility that an object
is chosen many times, the choice can be represented by a boolean sequence of . k
zeros and .n<> −<> 1ones, where zeros between two consecutive positions . iand . i<> +<> 1
(also before the ﬁrst one, or after the last one) express the multiplicity of the object
of type. i. A boolean sequence of.n<> +<> k<> −<> 1 positions is completely determined when
we chose the . kpositions where put zeros, therefore the number of these choices is
given by (the roles of objects and cells are exchanged, with respect to partitions):
.
(
n<> +<> k<> −<> 1
k
)
=<> n<>!
k
/<>k<>!
The following proposition is an important characterization of binomial coefﬁ-
cients. Its proof follows a general schema, which is common to a great number of
combinatorial formulas where the cases to count are partitioned into two distinct
subsets.
Proposition 2.1 <> The binomial coefﬁcient can be computed by means of the Tartaglia- Pascal equation:
.
(
n<> +<> 1
k<> +<> 1
)
=
(
n
k
)
+
(
n
k<> +<> 1
)
Proof <> Let us consider an object, denoted by . a0, among the given .n<> +<> 1objects.
Then .k<> +<> 1-sets of .n<> +<> 1elements can include .a0or not. Therefore, we can count
separately the number.Ck<>+<>1<>,<>n(<>a0)of.k<> +<> 1-sets of.n<> +<> 1elements including.a0and the
number.Ck<>+<>1<>,<>n(<>¬<>a0)of.k<> +<> 1-sets of.n<> +<> 1elements not including. a0. The number
.Ck<>+<>1<>,<>n(<>a0)is given by.
(
n
k
), because being.a0included, any.k<> +<> 1-set corresponds to
the choice of a .k-set over . nelements. The number .Ck<>+<>1<>,<>n(<>¬<>a0)is equal to .
(
n
k<>+<>1
),
because being .a0not included, it can be removed from the set of .n<> +<> 1objects,
which becomes an. nset, giving that.Ck<>+<>1<>,<>n(<>¬<>a0)<> =
(
n
k<>+<>1
)
. In conclusion, the number
of.
(
n<>+<>1
k<>+<>1
)is the sum of.
(
n
k
)and.
(
n
k<>+<>1
).
2.2 Strings and Rewriting
Strings are the mathematical concept corresponding to the notion of words as linear arrangements of symbols. Dictionaries are ﬁnite sets of strings. These two notions are very important in the analysis of strings, and present a very rich catalogue of
concepts and relations, in a great variety of situations.

2.2 Strings and Rewriting 13
Let us recall basic concepts and notation on strings (for classical textbooks, see
for example [ 8]). Strings are ﬁnite sequences over an alphabet.
Often strings are written as words, that is, without parentheses and commas or
spaces between symbols:.a1a2...<>an.When single elements are not relevant, generic
strings are often denoted by Greek letters (possibly with subscripts). In particular,
. λdenotes the empty string. The length of a string . αis denoted by .|<>α<>|(.|<>λ<>|=<>0),
whereas.α<>[<>i<>]denotes the symbol occurring at position. iof. α, and.α<>[<>i<>,<> j<>]denotes the
substring of . αstarting at position . iand ending at position . j(the symbols between
these positions are in the order they have in. α).
A string can be also seen as a distribution when any symbol of the alphabet is
associated with all the positions where it occurs in the string. Given an ordering
.a<> <<> b<> <<> ···<><<> z over the symbols of a string, the ordered normalization of a string
. αis the string
. a
|<>α<>|a
b
|<>α<>|b
...<>z
|<>α<>|z
where.|<>α<>|xis the number of times symbol. xoccurs in string. αand exponents denote
the concatenation of copies of the symbol (a number of copies equal to the value of
the exponent).
The set of all substrings of a given string. αis deﬁned by:
. sub<>(α)<> ={<>α<>[<>i<>,<> j<>]|<>1<> ≤<> i<> ≤<> j<> ≤|<>α<>|}.
and preﬁxes and sufﬁxes are given by:
. pref<>(α)<> ={<>α<>[<>1<>,<> j<>]|<>1<> ≤<> j<> ≤|<>α<>|}
. suff<>(α)<> ={<>α<>[<>i<>,<> |<>α<>|]|<>1<> ≤<> i<> ≤|<>α<>|}.
The most important operation over strings are:
1. concatenation <> of .α, β, usually denoted by the juxtaposition .αβThe <> overlap
concatenation <> of two strings .αγ, γβ,is.αγβwhere. γis the maximum substring
that is the sufﬁx of the ﬁrst string and the preﬁx of the second one;
2. length <> usually denoted by the absolute value notation.|<>α<>|;
3. preﬁx.α<>[<>1<>,<> j<>], with.1<> ≤<> j<> ≤|<>α<>| ;
4. sufﬁx.α<>[<> j<>,<> |<>α<>|], with.1<> ≤<> j<> ≤|<>α<>| ;
5. substitution.substfwith respect to a function. ffrom the alphabet of. αto another
(possibly the same) alphabet, where.substf(α)[<> j<>]=<> f<> (α<>[<> j<>]<>) for.0<> <<> j<> ≤|<>α<>| ;
6. reverse.rev<>(α), such that.rev<>(α)[<> j<>]=<>α<>[<>n<> −<> j<> +<> 1<>] ;
7. empty string. λ, such that, for every string. α:.λα<> =<> αλ<> =<> α . This string is very
useful in expressing properties of strings and can be seen as a constant or nullary
operation.

14 2 Basic Concepts
We remark the difference between <> substring <> and <> subsequence<>. Both notions
select symbols occurring in a sequence in the order they are, but a subsequence does
not assume that the chosen symbols are contiguous, while a substring of. αis always
of type.α<>[<>i<>,<> j<>]and selects all the symbols occurring in. αbetween the two speciﬁed
positions (included). The number of all possible substrings of a string of length . n
has a quadratic order, whereas the number of all possible subsequences is. 2
n
.
A class of strings over an alphabet .Ais called a (formal) <> language <> (over the
alphabet). Many operations are deﬁned over languages: the set-theoretic operations (union, intersection, difference, cartesian product) and more speciﬁc operations as concatenation
.L1L2(of two languages), iteration.L
n
(.n<> ∈<> N), and Kleene star.L
∗
:
. L1L2={<>αβ<> |<> α<> ∈<> L 1,β<> ∈<> L2}
. L
0
={<>λ<>}
. L
n<>+<>1
=<> LL
n
. L
∗
=
∪
i<> ∈<>N
L
i
.
Algorithms generating languages are called <> grammars<>. Automata are algorithms
recognizing languages or computing functions over strings. Grammars are intrin-
sically non-deterministic while recognizing automata can be deterministic or non-
deterministic. In the second case at each step many different consecutive steps are
possible, and one of them is non-deterministically chosen. A string is recognized by
such automata if for one of the possible computations the string is recognized.
We recall that a <> Turing Machine .M[ 9] can be completely described by means of
rewriting rules that specify as a tape conﬁguration can be transformed into another tape conﬁguration. A Turing conﬁguration is a string of type:
. α<>qx<>β
where. αis the string on the left of current position of.Mtape,. βis the string on the
right of this position,. qis the current state, and. xis the symbol of the tape cell that
.Mis reading. In this way, an instruction of .Msuch as .qxyq
'
R(in the state . qand
reading . xrewrite . xas . yand pass to state .q
'
moving on the cell to the right of the
current cell) can be expressed by the following rewriting rule on conﬁgurations:
. α<>qx<>β<> →<> α<>yq
'
β
whereas an instruction of .Msuch as .qxyq
'
L(in the state . qand reading . xrewrite
. xas . yand pass to state .q
'
moving on the cell to the left of the current cell) can be
expressed by the following two rewriting rules on conﬁgurations:
.α<>qx<>β<> →<> α<>q
'
∗
y<>β

2.2 Strings and Rewriting 15
. α<>xq
'
∗
β<> →<> α<>q
'
x<>β.
The representation above of a Turing machine implies a very important property
of the string rewriting mechanism: any process of string rewriting can be expressed
by means of rules involving at most two consecutive symbols. In fact, the rules of
the representation above involve the symbol of the state and only one symbol to
the right or to the left of it. Moreover, according to the Church-Turing thesis, any
computation can be realized by a suitable Turing machine, therefore any computation
can be described by 2-symbol rewriting rules. This is a well-known result in formal
language theory (Kuroda normal form representation).
Universal Turing machines. Uexist that verify the following universality equation:
. U<>(<<> M<>,α >)<>=<> M<>(α)
where.M<>(α)denotes the computation of Turing machine.Mwith the input string. α.
The angular parentheses .<>denote an encoding of the pair given by machine . M
(its instructions) and input string. α. Equality means that the left computation halts if
and only if the right computation halts, and moreover, their ﬁnal conﬁgurations are
essentially equal (being in a 1-to-1 correspondence). In 1956 [ 10] Shannon found a
universal Turing machine [ 9,11] working with only two symbols, say. 1and.B(the
symbol. Bfor blank).
An important class of grammars are the <> Chomsky Grammars <> essentially given
by a set of rewriting rules.α<> →<> β, where. α<> ∈<> A
∗
−<> T
∗
,β<> ∈<> A
∗
,<> S<> ∈<> A<> −<> T<>,<> T<> ⊂<> A
(. Tis called the terminal alphabet), and.⇒∗is the rewriting relation between a string
and any other string obtained by a chain of substring replacements where a left side
of a rule of a grammar .Gis replaced by its corresponding right side. The language
determined by.Gis given by:
. L<>(<>G<>)<> ={<>γ<> ∈<> T
∗
|<>S<> ⇒∗γ<> }<>.
A Chomsky grammar is of type 0 if no restriction is given to its rules. Chomsky grammars of type 0 generate all the recursively enumerable (or semi- decidable) languages, which belong to the class
.RE. For any language .L<> ∈<> REan
algorithm exists that provides a positive answer to the question.α<> ∈<> Lif and only if
.α<> ∈<> L. The class.RECof recursive (or decidable) languages is a proper subclass of
.REsuch that, for any.L<> ∈<> RECan algorithm exists that provides a positive answer
to the question .α<> ∈<> Lif .α<> ∈<> Land a negative answer if .α/<>∈<> L(proper inclusion
.REC<> ⊂<> RE was one of the most important achievements of the epochal paper of
Alan Turing, in 1936 [ 9]).
Recursive enumerable languages coincide with the domains or codomains of
Turing’s computable functions.

16 2 Basic Concepts
2.3 Variables and Distributions
Given a variable. X, we denote by. ∃Xits set of variability or its range. It is important to
remark that the set. ∃Xof elements where. Xtakes values can be completely unknown,
or not relevant.
When a probability is associated to a set of values assumed by variable . X, then
.Xis a random variable, which identiﬁes a probability distribution, where the whole
unitary probability is distributed among the subsets of. ∃X.
The notion of distribution is very general and is a key concept in mathematical
analysis, measure theory, probability, statistics, and information theory. Intuitively,
a distribution speciﬁes as a quantity is distributed in parts among the points or the
regions of a space. A very simple case of a <> ﬁnite distribution <> is a multiset of objects
where a number .mof occurrences is distributed among . kobjects where .n1are the
occurrences of the ﬁrst object, .n2those of the second one, and so on, up to . nk,the
occurrences of the last object, and:
.
∑
i<> =<>1<>...<>k
ni=<> m<>.
A discrete probability distribution is a family of real numbers. pi, indexed in a set. I
such that:
.
∑
i<> ∈<>I
pi=<> 1<>.
2.4 Events and Probability
The theory of probability emerged in the 17th century, with some anticipations by the Italian mathematician Girolamo Cardano (1501–1576) (<>Liber de ludo aleae<>),
Galileo (1564–1642) (<>Sopra le scoperte dei dadi<>, that is, About discoveries on dice),
and Christian Huygens (1629–1695) (<>De ratiociniis in ludo aleae<>). The basic rules for
computing probabilities were developed by Pascal (1623–1662) and Fermat (1601–
1665). The ﬁrst treatise on this subject was the book <> Ars Conjectandi <> by Jacob
Bernoulli (1654–1705), where binomial coefﬁcients appear in the probability of urn
extractions, and the ﬁrst law of large numbers is enunciated (<>Theorema Aureus<>:
frequencies approximate to probabilities in a large space of events). The idea of
events to which degrees of possibility are assigned is a change of perspective where
facts, as they are (<>modus essendi<>), are replaced by facts as they are judged (<>modus
conjectandi<>).
After Bernoulli’s work, De Moivre (1667–1754) and Laplace (1749–1827), found
the normal distribution as a curve for approximating the binomial distribution.
Bayes (1702–1752) discovered the theorem named by his name (independently
discovered also by Laplace) ruling the <> inversion of conditional probabilities<>. Gauss

2.4 Events and Probability 17
(1777–1855) recognized the normal distribution as the law of error distribution, and
Poisson (1781–1840) introduced his distribution ruling the law of rare events.
An <> event <> can be expressed by means of a proposition asserting that a variable. X
assumes a value belonging to a given subset. This means that the usual propositional operations
.¬<>,<> ∧<>,<> ∨(<>not, and, or<>) are deﬁned on events. A space of events is a special
boolean algebra (with 0, 1, sum, product, and negation)
A probability measure can be easily assigned to an event, as a value in the real
interval [0, 1], which can be seen as a sort of evaluation of the possibility that the
event has of happening.
The theory of probability concerns two different aspects:
(i) the study of probabilistic laws held in spaces of events,
(ii) the determination of the space of events which is more appropriate in a given
context.
The ﬁrst aspect constitutes a conceptual framework which is often independent
of the speciﬁc spaces of events. A comparison may help to distinguish the two aspects. Calculus and the theory of differential equations provide rules and methods
for solving and analyzing differential equations, but the choice of the right equation
which is the best description of a physical phenomenon is a different thing, which
pertains to the ability to correctly model phenomena of a certain type.
The axiomatic approach in probability theory was initiated by Andrej Nikolaeviˇ<>c
Kolmogorov (1903–1987) and was developed by the Russian mathematical school (Andrey Markov, Pafnuty Chebyshev, Aleksandr Yakovlevich Khinchin). It is com- parable to the axiomatic approach in geometry, and it is important for understanding
the logical basis of probability. From the mathematical point of view, probability
theory is part of a general ﬁeld of mathematics, referred to as <> Measure theory<>,ini-
tiated by French mathematicians of the last century (Émile Borel, Henri Lebesgue,
Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet).
Basic rules of probability
The <> conditional probability <> of an event . A, given an event. B, is denoted by.P<>(<>A<>|<>B<>).
It expresses the probability of .Aunder the assumption that event .Bhas occurred.
Formally (propositional connectives or set-theoretic operations are used on events):
. P<>(<>A<>|<>B<>)<> =
P<>(<>A<> ∧<> B<>)
P<>(<>B<>)
.
Events .Aand .Baresaidtobe <>independent<>, and we write .A<>||<>B,if . P<>(<>A<>|<>B<>)<> =
P<>(<>A<>). Events .Aand .Bare <> disjoint <> if .P<>(<>A<> ∧<> B<>)<> =<> 0 . The following rules connect
propositional operations to probabilities. Proposition .¬<>Ahas to be considered in
terms of complementary set, that is if.A<> =<> (<>X<> ∈<> S<>) , then.¬<>A<> =<> (<>X<> ∈ ∃X<> −<> S<>).

18 2 Basic Concepts
1. . P<>(<>A<>)<> ≥<> 0
2. . P<>(<>A<> ∨¬<>A<>)<> =<> 1
3. . P<>(<>A<> ∧¬<>A<>)<> =<> 0
4. . P<>(<>¬<>A<>)<> =<> 1<> −<> P<>(<>A<>)
5. . P<>(<>A<> ∨<> B<>)<> =<> P<>(<>A<>)<> +<> P<>(<>B<>)<> −<> P<>(<>A<> ∧<> B<>)
6. . P<>(<>A<>|<>B<>)<> =<> P<>(<>A<> ∧<> B<>)/<>P<>(<>B<>)
7. .A<>||<>B<> ⇔<> P<>(<>A<> ∧<> B<>)<> =<> P<>(<>A<>)<>P<>(<>B<>) .
The theory of probability is the ﬁeld where even professional mathematicians can
be easily wrong, and very often reasoning under probabilistic hypotheses is very
slippery. Let us consider the following examples from [ 12]. A military pilot has a
.2%chance of being shot down in each military ﬂight. What is the probability to die
in ﬁfty ﬂights? We might guess that he is almost sure to die. But this is completely wrong. The probability of dying is the sum of the probabilities of dying at the ﬁrst, at the second, and so on, up to the probability of dying at the last mission. Let us
denote by
. pthe probability of dying, then the sum of probabilities in all the ﬂights
is (the ﬂight. iis possible only if the pilot survives in the preceding ﬂights):
. p<> +<> (<>1<> −<> p<>)<>p<> +<> (<>1<> −<> p<>)
2
p<> +<> ...<>+<> (<>1<> −<> p<>)
49
p
A shorter evaluation of this probability (completely equivalent to the equation
above) is the probability of surviving.0<>.<>98
50
subtracted to 1. Therefore, the probability
of dying is.1<> −<> 0<>.<>98
50
=<> 0<>.<>64, which is very different from. 1.
A similar error was the origin of the problem posed by the <> Chevalier de Méré
to Pascal, about a popular dice game: <> Why the probability of one ace, rolling one
die 4 times, is greater than that of both aces, rolling two dice 24 times? <> In fact,
the probability of one ace is .1<>/<>6, and .4<>/<>6<> =<> 2<>/<>3 , analogously the probability of 2
aces is .1<>/<>36, and .24<>/<>36<> =<> 2<>/<>3 , so we may suppose that the two events: “1 ace in
4 rolls”, and “2 aces in 24 double rolls” are equiprobable. But the empirical evi- dence reported by Chevalier de Méré was against this conclusion. Namely, Pascal
(discussing with Fermat) solved the apparent paradox. In the ﬁrst game, P(no-ace-
in-4-tolls) =
.(<>5<>/<>6<>)
4
, therefore P(ace-in-4-rolls)= .1<> −<> (<>5<>/<>6<>)
4
=<> 0<>.<>5177. In the sec-
ond game, P(no-2-aces-in-24-double-rolls)= .(<>35<>/<>36<>)
24
, therefore P(2-aces-in-24-
double-rolls) = .1<> −<> (<>35<>/<>36<>)
24
=<> 0<>.<>4914. The simple mistake, as in the case of the
military pilot, is due to the sum of probabilities that are not disjoint (by ignoring rule
5 given above).
Bayes’ theorem
Cases of wrong probability evaluations are very frequent in contexts where condi-
tional probabilities are present. Bayes’ theorem, explains the nature of this difﬁculty
because it establishes the rule for inverting conditional probabilities.
In a simpliﬁed form, Bayes’ theorem asserts the following equation:

2.4 Events and Probability 19
Proposition 2.2
. P<>(<>A<>|<>B<>)<> =<> P<>(<>A<>)<>P<>(<>B<>|<>A<>)/<>P<>(<>B<>).
Proof <> By the deﬁnition of conditional probability, we have:
. P<>(<>A<> ∧<> B<>)<> =<> P<>(<>A<>|<>B<>)<>P<>(<>B<>)
analogously:
. P<>(<>A<> ∧<> B<>)<> =<> P<>(<>B<> ∧<> A<>)<> =<> P<>(<>B<>|<>A<>)<>P<>(<>A<>)
therefore, by equating the right members of the two equations above we obtain:
. P<>(<>A<>|<>B<>)<>P<>(<>B<>)<> =<> P<>(<>B<>|<>A<>)<>P<>(<>A<>)
from which the statement claimed by the theorem easily follows.<> □
Despite the simplicity of this proof, what the theorem asserts is an inversion of
conditions. In fact, .P<>(<>A<>|<>B<>)can be computed by means of .P<>(<>B<>|<>A<>). Assume that a
test .Tfor a disease .Dis wrong only in one out of 1000 cases. When a person is
positive on this test, what is the probability of having D? Is it .1<> −<> 0<>.<>001<> =<> 0<>.<>999 ?
No. In fact, this value confuses .P<>(<>D<>|<>T<> )with .P<>(<>T<> |<>D<>). The right way to reason is
the application of Bayes’ theorem; let us assume to know that .Daffects one out of
10000 persons, so.P<>(<>D<>)<> =<> 1<>/<>10000 , and that. Tis very reliable, with.P<>(<>T<> |<>D<>)<> =<> 1 .
The probability.P<>(<>T<> )is.11<>/<>10000because. Tis wrong from 1 to 1000, and in only
one out of 10000 persons with the disease D, test .Tis positive. In conclusion, the
probability of having.Dis less than 10%. In fact:
. P<>(<>D<>|<>T<> )<> =<> P<>(<>D<>)<>P<>(<>T<> |<>D<>)/<>P<>(<>T<> )<> =<> 1<>/<>10000<> ×<> 1<>/(<>11<>/<>10000<>)<> =<> 1<>/<>11<>.
Statistical indexes and Chebicev’s inequality
Let .μthe <> mean <> of a random variable .Xassuming real values .x1,<> x2,...<>xnwith
probabilities.p1,<> p2,...<>pnrespectively, then. μ, denoted also by.E<>(<>X<>), is deﬁned by
the following equation:
. E<>(<>X<>)<> =
∑
i
pixi
the second order moment of.Xis given by:
.E<>(<>X
2
)<> =
∑
i
pix
2
i

20 2 Basic Concepts
and in an analogous way, moments of higher orders can be deﬁned. the <> variance <> of
.Xis given by the second-order momentum of the deviation from the mean, that is:
. Var<>(<>X<>)<> =<> E<>[<>(<>X<> −<> μ)
2
)<>]=
∑
i
pi(<>xi−<> μ)
2
.
The following equation holds:
. var<>(<>X<>)<> =<> E<>(<>X
2
)<> −[<>E<>(<>X<>)<>]
2
.
In fact, from deﬁnitions, we obtain that:
.
∑
i
pi(<>xi−<> μ)
2
=
∑
i
pix
2
i
+
∑
i
piμ
2
−<> 2<>μ
∑
i
pixi
=<> E<>(<>X
2
)<> +<> μ
2
−<> 2<>μ
2
=<> E<>(<>X
2
)<> −[<>E<>(<>X<>)<>]
2
.
The standard deviation of. X,.sd<>(<>X<>)is deﬁned by:
. sd<>(<>X<>)<> =
√
Var<>(<>X<>).
The following theorem states a basic constraint of any statistical distribution: <> given
a random variable . X, the probability that .Xassume a value differing, in absolute
value, more than. tfrom the mean. μis less than.t
−<>2
times the variance of. X.
Proposition 2.3 <> (Chebichev Theorem)
. P<>(<>|<>X<> −<> μ<>|<> ><> t<>)<> ≤<> t
−<>2
E<>[<>(<>X<> −<> μ)
2
]
Proof
. P<>(<>|<>X<>|<> ><> t<>)<> ≤<> t
−<>2
E<>(<>X
2
)
In fact:
.
∑
|<>x<> |≥<>t
p<>(<>x<>)<> ≤
∑
|<>x<> |≥<>t
p<>(<>x<>)
x
2
t
2
≤<> t
−<>2
E<>(<>X
2
)
whence, replacing.Xby.(<>X<> −<> μ)the thesis holds.<> □
A direct consequence of the theorem above is the so-called <> Law of large numbers
in its weak form, originally proved by Bernoulli, stating that in a boolean sequence
of successes and failures, the frequency of successes rends to the success probability
according to which the sequence is generated. In formal terms. let.S<>(αn)be a boolean
sequence generated by a . nBernoulli boolean variables .X1,<> X2,...,<>X nall with a
probability of success (that is of having value 1) equal to . p.Let .S<>(αn)<> =<> Snthe

2.4 Events and Probability 21
number of successes in. αn,.Sn=<> X1+<> X2,...,<>+<>X nand.Ωnthe set of all sequences
of length. n. The following limit holds.
Proposition 2.4
. ∀<>ε<> ∈<> R
+
:<> lim
n<>→∞
P<>{<>αn∈<> Ωn:|<>Sn/<>n<> −<> p<>|<> >ε<>}=<>0
Proof <> The mean of a Bernoulli boolean variable .Xis . E<>(<>X<>)<> =<> 1<> ×<> p<> +<> 0<> ×<> (<>1<> −
p<>)<> =<> p. Therefore being.Snthe sum of. nindependent variables.E<>(<>Sn)<> =<> np. Anal-
ogously, the square deviation of any boolean random variable.Xis (.q<> =<> 1<> −<> p ):
. (<>1<> −<> p<>)
2
p<> +<> (<>0<> −<> p<>)
2
q<> =<> q
2
p<> +<> p
2
q<> =<> qp<>(<>q<> +<> p<>)<> =<> qp
therefore the sum of. nindependent boolean variables with success probability. phas
square deviation.npq<> =<> np<>(<>1<> −<> p<>) .Fix. ε, then by Chebichev’s inequality:
. P<>{<>αn∈<> Ωn:|
S
n
n
−<> E<>(<>S
n)<>|<> >ε<>}≤
Var<>(<>S
n)
ε
2
.
The mean value and the variance of.
Sn
n
are:
. E
(
S
n
n
)
=
E<>(<>S
n)n
=<> p
. Var
(
S
n
n
)
=
Var<>(<>S
n)
n
2
=
np<>(<>1<> −<> p<>)
n
2
=
p<>(<>1<> −<> p<>)
n
thus, we obtain:
. P<>{<>αn∈<> Ωn:|
S
n
n
−<> p<>|<> >ε<>}≤
p<>(<>1<> −<> p<>)
n<>ε
2
so, passing to the limit:
. lim
n<>→∞
P<>{<>αn∈<> Ωn:|
S
n
n
−<> p<>|<> >ε<>}≤0
but the probability cannot be negative, therefore:
.lim
n<>→∞
P<>{<>αn∈<> Ωn:|
S
n
n
−<> p<>|<> >ε<>}=<>0<>.
□
Besides, mean, momenta, variance, and standard deviation, the <> correlation coefﬁ-
cient <> is an important index for expressing the degree of reciprocal inﬂuence between
two variables, on the basis of the distributions of their values. Correlation is based on
the notion of <> covariance <> that is expressed by the mean of the product of deviations
(. μand. νare the means of.Xand. Y, respectively):

22 2 Basic Concepts
. cov<>(<>X<>,<> Y<> )<> =<> E<>[<>(<>X<> −<> μ)<>[<>Y<> −<> ν)<>]<>.
Then, the correlation coefﬁcient.ρ(<>X<>,<> Y<> )is given by (assuming a ﬁnite variance
for.Xand. Y):
. ρ(<>X<>,<> Y<> )<> =
cov<>(<>X<>,<> Y<> )
Var<>(<>X<>)<>Var<>(<>Y<> )
.
The intuition concerning covariance is very simple. If two variables deviate from
their means in a coherent way they are related, and the product of the deviations
is positive when their deviations agree and is negative otherwise. Moreover, the
agreements and disagreements are reasonably weighted by the probabilities with
which they occur. The denominator in the correlation ratio is introduced to normalize
the covariance with respect to the variances of the two variables.
References
1. Manca, V.: Il Paradiso di Cantor. La costruzione del linguaggio matematico, Edizioni Nuova
Cultura (2022)
2. Manca, V., Arithmoi. Racconto di numeri e concetti matematici fondamentali (to appear)
3. Fränkel, A.A.: Set Theory and Logic. Addison-Wesley Publishing Company (1966)
4. Halmos, P.: Axiomatic Set Theory. North-Holland (1964)
5. Manca, V.: Formal Logic. In: Webster, J.G. (ed.) Encyclopedia of Electrical and Electronics Engineering, vol. 7, pp. 675–687. Wiley, New York (1999)
6. Manca, V.: Logical string rewriting. Theor. Comput. Sci. <> 264, 25–51 (2001)
7. Aigner, M.: Discrete Mathematics. American Mathematical Society, Providence, Rhode Island (2007)
8. Rozenberg, G., Salomaa, A.: Handbook of Formal Languages: Beyonds words, vol. 3. Springer, Berlin (1997)
9. Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. <> 42<>(1), 230–265 (1936)
10. Shannon, C.E.: A universal Turing machine with two internal states. In: Automata Studies, Annals of Mathematics Studies, vol. 34, pp. 157–165. Princeton University Press (1956)
11. Minsky, M.L.: Computation: Finite and Inﬁnite Machines. Prentice-Hall Inc. Englewood Cliffs, N. J. (1967)
12. Aczel, A.D.: Chance. Thunder’s Mouth Press, New York (2004)

Chapter 3
Information Theory
Introduction
Information theory “ofﬁcially” begins with Shannon’s booklet [ 1] published in 1948.
The main idea of Shannon is linking information with probability. In fact, the starting
deﬁnition of this seminal work is that of <> information source <> as a pair .(<>X<>,<> p<>), where. X
is a ﬁnite set of objects (data, signals, words) and. pis a probability function assigning
to every.x<> ∈<> Xthe probability.p<>(<>x<>)of occurrence (emission, reception, production).
The perspective of this approach is the mathematical analysis of communication
processes, but its impact is completely general and expresses the probabilistic nature
of the information. Information is an inverse function of probability. because it is a
sort of a posteriori counterpart of the a priori uncertainty represented by probability,
measuring the gain of knowledge when an event occurs. For this reason, the more an
event is rare, the more it is informative. However, if event
.Ehas probability.pE,for
several technical reason it is better to deﬁne.inf<>(<>E<>)as.lg<>(1/<>pE)<> =−<>lg(<>p E)rather
than .1/<>pE. In fact, the logarithm guarantees the information additivity for a joint
event .(<>E<>,<> E
'
)where components are independent, giving . inf<>(<>E<>,<> E
'
)<> =<> inf<>(<>E<>)<> +
inf<>(<>E
'
)
. However, it is important to remark that the relationship between information
and probability is in both verses because as Bayesian approaches make evident, information can change the probability evaluation (conditional probability, on which Bayes theorem is based, deﬁnes how probability changes when we know that an event
occurred). A famous example of this phenomenon is the three-doors (or Monty Hall)
dilemma [
2], which can be fully explained by using the Bayes theorem.
In this chapter, we give a quick overview of the basic concepts in Information
Theory. The reader is advised to refer to [ 1, 3, 4] for more details on Information
and Probability theories. Some speciﬁc concepts and facts about probability will be
more exhaustively considered in the next chapter.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
V. Manca and V. Bonnici, <> Infogenomics<>, Emergence, Complexity and Computation 48,
https://doi.org/10.1007/978-3-031-44501-9_3
23

24 3 Information Theory
3.1 From Physical to Informational Entropy
Shannon founded Information Theory [ 1, 3] on the notion of <> Information Source<>,
a pair .(<>A<>,<> p<>), given by a set of data .Aand a probability distribution .pdeﬁned
on . A, or even, a pair .(<>X<>,<> pX)of a variable .Xassuming values with an associated
probability distribution .pX(.Ais the set on which .Xassumes values). In this case,
only the variable. Xcan be indicated if.pXis implicitly understood. In this framework,
Shannon deﬁned a measure of <> information quantity <> of the event .X<> =<> a(. Xassumes
the value. a):
. Inf<>(<>X<> =<> a<>)<> =−<>lg
2
(<>pX(<>a<>))
where.pX(<>a<>)is the probability of the event.(<>X<> =<> a<>). The intuitive motivation for the
equation above is that the information quantity associated with an event is inversely
proportional to the probability of the event, and moreover, the information quantity
has to be additive for pairs of independent events (.p<>(<>a1,<> a2)<> =<> p<>(<>a 1)<>p<>(<>a2)):
. Inf<>(<>E1,<> E2)<> =<> Inf<>(<>E 1)<> +<> Inf<>(<>E 2).
On the basis of this deﬁnition, if. ΔXdenotes the range of the variable. X,the <>entropy
of the information source.(<>X<>,<> pX)is deﬁned by:
. H<>(<>X<>,<> pX)<> =
∑
a<>∈ΔX
pX(<>a<>)<>Inf<>(<>X<> =<> a<>)<> =−
∑
a<>∈ΔX
pX(<>a<>)<> lg
2
pX(<>a<>).
Therefore, the entropy of an information source is the mean (in a probabilistic sense)
of the information quantity of the events generated by the information source.(<>X<>,<> pX).
One crucial result about entropy is the <> Equipartition Property <> (proved by Shan-
non in an appendix to his booklet): in the class of variables assuming the same value of
.Xthe value of the entropy reaches its maximum with a source .(<>Y<>,<> qY)where . qY
is the uniform probability distribution (all the values of variable. Yare assumed with
the same values).
Let us recall that Shannon’s idea has very ancient historical roots (probably
unknown to Shannon). The encoding method “Caesar” (used by Julius Caesar) is a
very simple way of encoding messages for hiding their contents so that only whoever
knows a deciphering key can access to them. This method is based on a one-to-one
function assigning a letter.f<> (<>X<>)to a letter. X. Given a text, if you replace each letter
by using this function, then the message becomes unreadable unless you use the inverse of
. ffor recovering the original message. In the eightieth century, the Ara-
bic mathematician Al-Kindi discovered a method for cracking “Caesar” encoding,
entirely based on frequencies. Let us assume to know the language of the original
messages, and to collect a great number of encrypted messages. In any language,
the frequency of each letter is almost constant (especially if it is evaluated for long
texts). Then, if we compute the frequencies of letters in the encrypted messages, it
results that, with a very good approximation, we can guess, for every letter of the

3.1 From Physical to Informational Entropy 25
alphabet which is the letter in the encrypted texts having a similar frequency. This
letter is, with a good probability the letter corresponding to the letter of plain texts,
in the encrypted messages. In this way, the deciphering key. fcan be discovered and
messages can be disclosed.
The consequence of this deciphering method is that the essence of any symbol in
a symbolic system is intrinsically related to its frequency, which is a special case of
probability (an empirical probability). For this reason, information is directly related
to probability.
The notion of informational entropy is strongly related to thermodynamic entropy
. S, which emerged in physics since Carnot’s analysis of heat-work conversion [ 5] and
was named by Clausius “entropy” (from Greek en-tropos meaning “internal verse”).
Physical entropy is subjected to a famous inequality stating the <> Second Principle of
Thermodynamics <> for isolated systems (systems that do not exchange energy with
their environment), entropy cannot decrease in time:
. Δ<>S<> ≥<> 0<>.
In years 1870s, Ludwig Boltzmann started a rigorous mathematical analysis of a
thermodynamic system consisting of ideal gas within a volume, aimed at explaining
the apparent contradiction of the inequality above with respect to the equations of
Newtonian mechanics, underlying the microscopic reality on which heat phenomena
are based on. More precisely, Boltzmann’s question was: “From where the inequality
comes from, if molecules colliding in a gas follow mechanics equational laws with
no intrinsic time arrow?”
For answering the above question, Boltzmann introduced, in a systematic way, a
probabilistic perspective in the microscopic analysis of physical phenomena. For its
importance, his approach transcends the particular ﬁeld of his investigation. In fact,
after Boltzmann, probability and statistics [ 4, 6, 7] became crucial aspects of any
modern physical theory. Boltzmann deﬁned a function . H, which in discrete terms
can be expressed by:
. H<> =
∑
i<> =1,<>n
nilg
2
ni
where.niare the number of gas molecules having velocities in the.i-class of velocities
(velocities are partitioned in. ndisjoint intervals).
By simple algebraic passages, it turns out (details are given in Section 5.4) that
the .Hfunction coincides, apart from additive and multiplicative constants, with
Shannon’s entropy (Shannon quotes Boltzmann’s work). On the basis of.Hfunction
Boltzmann proved the microscopic representation of Clausius entropy. S[ 8].
.S<> =<> k<> log
e
W (3.1)
where.Wis the number of distinguishable micro-states associated with the thermo-
dynamic macro-state of a given system (two micro-states are indistinguishable if

26 3 Information Theory
the same velocity distribution is realized apart from the speciﬁc identities of single
molecules) and. kis the so-called <> Boltzmann constant<>.
However, despite this remarkable result, Boltzmann was not able to deduce that
.Δ<>S<> ≥<> 0,byusing <>H <> function. The so-called Theorem <> H <> stating .Ht<> +<>1≤<> Ht,or. Δ<>H<> ≤
0, was never proved by him in a satisfactory way (from.Δ<>H<> ≤<> 0inequality. Δ<>S<> ≥<> 0
follows as an easy consequence of the above equation .S<> =<> k<> log
e
W, where the
opposite verses of inequalities are due to the different signs of Boltzmann’s <> H <> and
Shannon’s <> H<>)[ 5, 9– 12].
The microscopic interpretation of thermodynamic entropy given by Boltzmann
has a very general nature expressed by the <> Entropy Circular Principle <> ECP, cor- responding to Boltzmann’s “Wahrscheinlichkeit” Principle. In terms of Shannon
information sources, ECP can be formulated by the following statement.
The entropy of a source.(<>X<>,<> p<>)is proportional to the logarithm of the number.Wof
different sources generating the same probability distribution. p, that is, the entropy
of a given source is proportional to the number of sources having that entropy.
Entropy Circular Principle <> (ECP)
. H<>(<>X<>,<> pX)<> =<> c<> lg
2
|<>X<>|
for some constant. c, where:
. X<> ={<>(<>Y<>,<> p Y)<> |ΔX<> =ΔY<> ,<> p X=<> pY}
From this principle will emerge an interesting link between probabilistic and dig-
ital aspects of information. In fact, it can be proved that.lg
2
ncorresponds essentially
to the minimum average number of binary digits necessary for encoding . ndiffer-
ent values (the exact value .digit2(<>n<>)is bounded by: . (<>[<>lg2(<>n<>)<>]−<>2<>)<<>digit 2(<>n<>)<
n<>(<>[<>lg
2(<>n<>)<>]<>)
). Therefore, entropy corresponds (apart from multiplicative constants)
to the average length of codewords encoding all the sources with the same entropy. This digital reading of the entropy is also related to the First Shannon Theorem,
stating that
.H<>(<>X<>,<> pX)is a lower bound for the (probabilistic) mean of yes-no ques-
tions necessary to guess a value .a<> ∈ΔXby asking .a<> ≤<> b<>?, for suitable values . bin
. ΔX. This explains in a very intuitive manner the meaning of.H<>(<>X<>,<> p<>), as the average
uncertainty of the source.(<>X<>,<> p<>).
In this sense, entropy is, at the same time, the average uncertainty of the events
generated by the source, and at the sane time, the average uncertainty in identifying the source, in the class of those with the same probability distribution. The uncertainty
internal <> to the space of events coincides with the <> external <> uncertainty of the event
space, among all possible event spaces.
In this perspective, the ECP principle unveils the intrinsic <> circularity <> of entropy,
because.H<>(<>X<>,<> pX)is completely determined by.pXbut, at the same time, corresponds
to the (logarithm of the number of) ways of realizing distribution .pXby random
variables assuming the same values of. X.
The reason for Boltzmann’s failure in correctly proving .Htheorem is due to
the fact that this theorem is, in its essence, an information Theory theorem (see [ 8,

3.2 Entropy and Computation 27
12] for proofs of .Htheorem). In fact, in simple words, the irreversible increase of
entropy in time, results from a collective effect of the large number of reversible
elementary events, according to the casual nature of molecular collisions. Molecules
that collide, exchanging information, produce, in time, an increasing uniformity of
molecular velocities. The global energy does not change (collisions are elastic),
but differences between velocities decrease and this provides a greater value of the
entropy, according to the entropy equipartition property.
3.2 Entropy and Computation
We showed how information entropy is related to physics and how time in physics is
related to entropy. In computations the informational and physical components are
intertwined, because any computational device, even when abstractly considered, is
based on a physical process, where states, symbols, or conﬁgurations are transformed
along a dynamics starting with some initial conﬁgurations, and eventually ending,
if computation reaches a result, with a ﬁnal conﬁguration (the initial conﬁguration
encodes input data, while the ﬁnal conﬁguration encodes the corresponding output
data).
In this perspective, computation is a trajectory in space of events, where events
can be conceived as suitable sets of states (according to micro-state/macro-state
Boltzmann’s distinction, any macro-state consists of a collection of micro-states).
On the other side, if computation is a process intended to acquire information, this
representation suggests that the conﬁgurations along a computation have to reduce
the number of internal micro-states. In fact, the uncertainty of a conﬁguration corre- sponds with the number of states it contains, and a computation tends to reduce the
initial uncertainty by reaching more certain conﬁgurations, which provide a solution,
by satisfying the constraints of the given problem.
In terms of entropy, if the number of possible states reduces along a computation,
then the entropy .lg<> W(.Wnumber of states) decreases along the computation. In
conclusion, according to the outlined perspective, the computation results to be an
anti-entropic process. For this reason, from a physical point of view, a computation
is a dissipative process releasing energy (heat) in the environment. A computational
device is comparable with a living organism: it increases the environment entropy
for maintaining his low internal entropy [
11], analogously, a computational device
increases the environment entropy for proceeding in the computation and providing
a ﬁnal result.
An intuitive way for realizing that “information is physical” is a simple device
called Centrifugal Governor<>, invented by James Watt in 1788 for controlling his
steam engine. The principle on which it is based is that of a “conical pendulum”. The
axis of a rotating engine is supplied with an attached rigid arm terminating with a mass
that can assume an angle with respect to the axis (two symmetrical arms in Watt’s
formulation, with a negligible mass at end of any arm). When the axis is rotating, in
proportion to the rotation speed, the mass at end of the arm is subjected to a centrifugal

28 3 Information Theory
Fig. 3.1 <> The schema of
Centrifugal Governor
forcerisingthe arm(seeFig. 3.1). This rising opens a valve, in proportion to the
vertical rising of the arm, which diminishes a physical parameter related to the speed
(pressure in the case of a steam engine), and consequently, decreases the rotation
speed. This phenomenon is a <> negative feedback <> that, according to the length of the
arm, stabilizes the rotation speed to a ﬁxed velocity (negative means that control
acts in the opposite verse of the controlled action). It realizes a kind of <> homeostasis
(keeping some variables within a ﬁxed sub-range of variability), a property that is
crucial for any complex system and thus typical of living organisms.
The term “cybernetics” introduced by Norbert Wiener [ 13] comes from a Greek
root expressing the action of guiding or controlling, and is essentially based on the
image of a Centrifugal Governor, where the arm length encodes a piece of information
capable of controlling the rotation speed (Wiener’s book title is: Cybernetics, or
Control and Communication in the Animal and the Machine). Information directs
processes, but it is realized by a physical process too. And in the centrifugal governor,
.mghis the energetic cost of the control exerted by the arm on the rotation (see Fig.
3.1:. mis the mass at the end of the arm,. gthe gravity acceleration, and. his the vertical
rising of the arm). We remark that from equation .m<>v
2
/2<> =<> mgh(kinetic energy =
potential energy) it follows that rising .h<> =<> v
2
/2<>gdoes not depend on the value of
the mass, but only on the speed (that has to be sufﬁcient to provide a centrifugal force
able to open the valve).
The investigation of the relationship between information and physics has a long
history, going back to the famous <> Maxwell demon, introduced by the great physicist
in addressing the problem of how an intelligent agent can interfere with physical
principles. His demon was apparently violating the second principle of thermody-
namics, for his ability to get information about the molecules’ velocity (but also this
acquisition of information requires an energetic cost). A very intensive and active line
of research was developed in this regard (see [
14] for a historical account of these
researches) and continues to be developed, even in recent years, especially under
the pressure of the new results in quantum information and in quantum computing, which show how much it is essential the informational perspective for a deep analysis
of quantum phenomena [
15].
It is becoming always more evident that any mathematical model of physical phe-
nomena is based on data coming from information sources generated by measurement
processes. This fact is not of secondary relevance, with respect to the construction

3.2 Entropy and Computation 29
Fig. 3.2 <> Landauer’s
minimal erasure as a
compression of the state
space: in <> a <> the ball can be
located in any of the two
parts 0, 1 of the volume; in <> b
it is conﬁned in part 1,
therefore in (<>b) the state
space is reduced from two
possibilities to only one of
them
of the model, especially when the data acquisition processes are very sophisticated
and cannot ingenuously be considered as mirrors of reality, rather, the only reality
on which reconstructions can be based is just the information source resulting from
the interactions between the observer and the observed phenomena.
In a celebrated paper [ 16] Rolf Landauer asserts the <> Erasure Principle, according
to which the erasure of a bit during a computation has an energetic cost of . kT<> ln 2
(. kBoltzmann constant, .Tabsolute temperature) [ 17]. This means that, if the com-
putation entropy diminishes (for the reduction of computation conﬁguration states),
then the environment entropy has to compensate this decrease by releasing a corre- sponding quantity of heat
.STin the environment.
The erasure principle is nothing else than a direct derivation of Boltzmann Equa-
tion 3.1, when from two possible states one of them is chosen, by passing from. W<> =<> 2
to.W<> =<> 1. However, it is important to remark that “erasing” has to be considered in
the wide sense of decreasing states of a computation conﬁguration, as indicated in Fig.
3.2 (the term erasure could be misleading, because it means writing a special
symbol on a Turing machine tape).
A continuation of Landauer’s research has been developed by Bennet and others
[ 14,18– 20](in [ 14] a historical account of the theory of reversible computation is
given). Computation steps can be reversible and irreversible. The ﬁrst kind of step
arise when information is not lost after the step and it is possible to go back by returning to the data previously available. The irreversible steps are those that do not satisfy the condition of reversibility because some data are lost after the step. In
this framework, it is shown that any computation can be performed in a reversible
way, by using suitable strategies, where all the steps of a computation ending in a
given conﬁguration are copied in a suitable zone of memory such that from the ﬁnal
state it is possible to go back in a reverse way. Of course, in this way computation is
reversible, but we are perplexed about the effective meaning of this result. In fact, if
the information is gained by reducing the set of states of the initial conﬁguration, then
in the case of a reversible computation no new information is obtained at end of the
computation. This would require a different way of considering computations, where
the evaluation of the amount of the gained information, at the end of computation,
has to be deﬁned in other terms. Otherwise, if a computation does not generate
information, for what reason it is performed? What is the advantage of obtaining a

30 3 Information Theory
result when computation halts? If a result is not new information, what gain is given
by it? The theory of reversible computation, even if correct, is surely incomplete, as
far as it does not answer these open questions [ 21].
Physical states representing data are a proper subset of all physical states of a
system. Those that represent data are selected by the observer of the system who is
using it for computing. But computation is not only a physical activity, it is mainly a coding-decoding activity according to a code chosen by another system coupled
with the computation. This means that in any computation we need to distinguish
between a passive and an active component. Forgetting this coupling and analyzing
computations only on the side of the operations applied to the tape (of a Turing
machine) can be useful, but does not tell the whole story of the computing process.
This partiality is the source of some conclusions drawn in the theory of reversible
computation. If even all the logical gates are reversible, the internal states of the
“interpreter” processor could forget an essential aspect of the computation. Therefore,
is not so obvious that we can always get rid of erasing operations, when also the
program determination is taken into account, rather than only data transformations.
The hidden trap, in analyzing a computation, is that of considering only the entropy
of data, by forgetting that an agent provides the input data and a program, and he/she
decodes the data obtained at end of the computation (if it halts). In other words, no computation can forget the agent performing it. Computation is not only in the computing device, but also in the interaction between an agent and a computation
device. For this reason, also the entropic/energetic evaluation of the process that in
the agent is coupled with the computing device has to be taken into account.
For example, in reversible computation theory, a comparison is often presented
with the RNA Polymerase transcription from DNA to RNA, by arguing that this
copy phenomenon is reversible. But in this comparison, no mention is done about the anti-entropic character of RNA Polymerase that, in order to be maintained “alive”,
requires some corresponding anti-entropic processes.
In conclusion, the theory of reversible computation is sure of great interest, by dis-
closing subtle relations among time, space, and energy [ 14], however, what it claims
raises several controversial aspects that need further clariﬁcations and experiments.
An anecdote reports that when Shannon asked John von Neumann to suggest him
a name for the quantity. S, then von Neumann promptly answered: “Entropy. This is
just entropy”, by adding that this name would have been successful because only a
few men knew exactly what entropy was.
Entropy has a paradoxical aspect, due to its intrinsic probabilistic nature. The
paradox is apparent at the beginning of Shannon’s paper [ 1]. In fact, Shannon says
that he is searching for a measure of information, or uncertainty. But how can we
consider these notions as equivalent? It would be something like searching for a
measure of knowledge or ignorance. Can we reasonably deﬁne the same measures
for opposite concepts? The answer to these questions can be found in the intrinsic
orientation of events in time. When an event
. ehappens with a probability . p,we
can measure its information by a function of its a priori uncertainty . p, but after
it happens, we gain, a posteriori, the same amount of information associated to
. p, therefore a priory uncertainty is transformed into information. The same kind

3.3 Entropic Concepts 31
of opposite orientation is always present in all informational concepts and it is
often a source of confusion when the perspective of considering them is not clearly
deﬁned [ 7].
As we will show, data can be always expressed by strings, that is, data can be
linearized over a given alphabet of symbols. This assumption is a prerequisite of
Shannon’s analysis that can be summarized by his three fundamental theorems.
First Shannon theorem <> provides a lower bound to the mean length of strings
representing the data emitted by an information source (mean calculated with respect
to the probability distribution of the source). More precisely, the entropy of the source, which is independent of any method of data representation, turns out to provide this
lower bound.
Second Shannon theorem <> is based on mutual information: <> even if data of an
information source are transmitted with some noise along a channel, it is possible
to encode them in such a way that transmission could become error-free, because
there exist transmission codes such that, the longer are the transmission encodings,
the more error probability approaches to zero.
In more precise terms, the theorem establishes quantitative notions giving the
possibility of avoiding error transmission when the <> transmission rate <> is lower than
the channel <> capacity of transmission, where these notions are formally deﬁned in suitable terms by using mutual information (between the <> transmitter <> information source and the bf receiver information source).
Third Shannon theorem <> concerns with signals. To this end, the entropic notions
are extended to the case of continuous information sources, and then, by using these
continuous notions, quantitative evaluations are proven about safe communications
by means of continuous signals.
3.3 Entropic Concepts
Entropy allows us to deﬁne other informational concepts further developing the
probabilistic approach.
Let us remark that the notion of information source is completely equivalent to
that of an aleatory or random variable, that is, a variable.Xto which is associated a
probability.pX, where.pX(<>x<>)is the probability that.Xassumes the value. x. In other
words, the random variable.Xidentiﬁes the information source.(<>A<>,<> pX), where. Ais
the set of values assumed by the variable. X. <> Viceversa, an information source. (<>A<>,<> p<>)
identiﬁes a random variable .Xthat assumes the values of .Awith the probability
distribution . p. For this reason, very often, entropy and other entropic concepts, are
equivalently given for information sources, random variables, and/or probability
distributions (discrete or continuous). In the following, according to the speciﬁc
context, we adopt one of these alternatives.

32 3 Information Theory
Conditional entropy
Given two random variables .Xe .Yof corresponding distributions .pXand .pY,the
joint variable.(<>X<>,<> Y<> )is that assuming as values the pairs.(<>x<>,<> y<>)of values assumed by
. Xand. Yrespectively, where.pXYis the joint distribution.pXY(<>x<>,<> y<>), shortly indicated
by .p<>(<>x<>,<> y<>), giving the probability of the composite event .X<> =<> xand .Y<> =<> y(when
arguable subscripts in.pX,<> pY,<> pXYwill be avoided).
The <> joint entropy.H<>(<>X<>,<> Y<> )of.(<>X<>,<> Y<> )is deﬁned by:
. H<>(<>X<>,<> Y<> )<> =−
∑
x<> ∈<> A
∑
y<>∈<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>x<>,<> y<>)
or:
. H<>(<>X<>,<> Y<> )<> =−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>x<>,<> y<>).
By deﬁnition of conditioned probability, we have that .p<>(<>y<>|<>x<>)<> =
p<>(<>x<> ,<>y<>)
p<>(<>x<> )
.Given two
random variables .Xe . Y, <> conditional entropy .H<>(<>Y<> |<>X<>) is deﬁned by the following
equation, where .E<>[<>...<>]denotes the expected value with respect to the distribution
probability of the random variable inside brackets:
.
H<>(<>Y<> |<>X<>)<> =−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>y<>|<>x<>)
H<>(<>Y<> |<>X<>)<> =−
∑
x<> ,<>y<>∈<> A<>×<>B
[<>p<>(<>x<>)<>p<>(<>y<>|<>x<>)<>]<> log<> p<>(<>y<>|<>x<>)
=−
∑
x<> ∈<> A
p<>(<>x<>)
∑
y<>∈<>B
p<>(<>y<>|<>x<>)<> log<> p<>(<>y<>|<>x<>)
=
∑
x<> ∈<> A
p<>(<>x<>)<>H<>(<>Y<> |<>X<> =<> x<>).
In the above deﬁnition, the joint probability is multiplied by the logarithm of the
conditional entropy (rather than, conditioned probability multiplied by its logarithm).
This apparent asymmetry is motivated by the following equation, which follows
from the deﬁnition above, where for any value
. xof. X,.Hxdenotes the entropy of. X
conditioned by the event.X<> =<> x:
. H<>(<>Y<> |<>X<>)<> =
∑
x<> ∈<> A
p<>(<>x<>)<>Hx.
The following proposition corresponds to the well-known relation between joint
and conditional probabilities:.p<>(<>x<>,<> y<>)<> =<> p<>(<>x<>)<>p<>(<>y<>|<>x<>).

3.3 Entropic Concepts 33
Proposition 3.1
. H<>(<>X<>,<> Y<> )<> =<> H<>(<>X<>)<> +<> H<>(<>Y<> |<>X<>).
Proof
.
H<>(<>X<>,<> Y<> )<> =−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>x<>,<> y<>)
=−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<>(<>p<>(<>x<>)<>p<>(<>y<>|<>x<>))
=−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>x<>)<> −
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>y<>|<>x<>)
=−
∑
x<> ,<>y<>∈<> A<>×<>B
p<>(<>x<>,<> y<>)<> log<> p<>(<>x<>)<> +<> H (Y<> |<>X)
=−
∑
x<> ∈<> A
·
∑
y<>∈<>B
(<>p(x<>,<> y)log<> p<>(<>x<>))<> +<> H (Y<> |<>X)
=−
∑
x<> ∈<> A
⎛
⎝
∑
y<>∈<>B
p(x<>,<> y)
⎞
⎠log<> p<>(<>x<>)<> +<> H (Y<> |<>X)
=−
∑
x<> ∈<> A
p<>(<>x<>)<> log<> p<>(<>x<>)<> +<> H (Y<> |<>X)
H<>(<>X<>,<> Y<> )<> =<> H<>(<>X<>)<> +<> H (Y<> |<>X)
Entropic divergence
Given two probability distributions .p<>,<> qtheir <> entropic divergence<>, .D<>(<>p<>,<> q<>),also
denoted by.KL<>(<>p<>,<> q<>) after Kullback and Leibler [ 3], is the probabilistic mean, with
respect to distribution . p, of the differences of information quantities associated to
distribution. pand. qrespectively:
.D<>(<>p<>,<> q<>)<> =
∑
p<>(<>x<>)<>[<>lg<> p<>(<>x<>)<> −<> lg<> q<>(<>x<>)<>] (3.2)
or, indicating by.Epthe mean with respect to the probability distribution. p:
.D<>(<>p<>,<> q<>)<> =<> E p[<>lg<> p<>(<>x<>)<> −<> lg<> q<>(<>x<>)<>] (3.3)
or, by using a property of logarithm:

34 3 Information Theory
. D<>(<>p<>,<> q<>)<> =<> E p
[
log
p<>(<>x<>)
q<>(<>x<>)
]
When.q<> =<> p,.D<>(<>p<>,<> p<>)<> =<> 0 . Moreover, equations.0log
0
q
=<> 0e.p<> log
p
0
=∞are
usually assumed.
Mutual Information
Let. pbe a probability distribution deﬁned over.X<> ×<> Y, where.pX,<> pYare the proba-
bilities of.X<>,<> Yrespectively, also called <> marginal probabilities<>.The <>mutual infor-
mation .I<> (<>X<>,<> Y<> )is given by the entropic divergence between distributions . p(<>X<>,<>Y<> )
and .pX×<> pY, where .pXYis the joint probability assigning to each pair .(<>x<>,<> y<>)the
probability .p<>(<>x<>,<> y<>)of the joint event .X<> =<> x<>,<> Y<> =<> y , while .pX×<> pYis the product
of the marginal probabilities, assigning to each pair.(<>x<>,<> y<>)the product.p<>(<>x<>)<>p<>(<>y<>) of
the two probabilities.pX,<> pY:
.I<> (<>X<>,<> Y<> )<> =<> D<>(<>p XY,<> pX×<> pY). (3.4)
By the deﬁnition of.I<> (<>X<>,<> Y<> )we have:
.I<> (<>X<>,<> Y<> )<> =
∑
x<> ∈<>X<>,<>y<>∈<>Y
p<>(<>x<>,<> y<>)<> lg<>[<>p<>(<>x<>,<> y<>)/<>p<>(<>x<>)<>p<>(<>y<>)<>]<>.
(3.5)
therefore:
.I<> (<>X<>,<> Y<> )<> =
∑
x<> ∈<>X<>,<>y<>∈<>Y
p<>(<>x<>,<> y<>)<>[<>lg<> p<>(<>x<>,<> y<>)/<>p<>(<>x<>)<>]−
∑
x<> ∈<>X<>,<>y<>∈<>Y
p<>(<>x<>,<> y<>)<> lg<> p<>(<>y<>)<>]
(3.6)
that is:
.I<> (<>X<>,<> Y<> )<> =
⎛
⎝
∑
x<> ∈<>X<>,<>y<>∈<>Y
p<>(<>x<>,<> y<>)<>[<>lg<> p<>(<>y<>|<>x<>)<>]
⎞
⎠−
∑
y<>∈<>Y
p<>(<>y<>)<> lg<> p<>(<>y<>)<>]
(3.7)
that can be written as:
.I<> (<>X<>,<> Y<> )<> =<> H<>(<>Y<> )<> −<> H<>(<>Y<> |<>X<>). (3.8)
If we consider the conditional entropy .H<>(<>Y<> |<>X<>) as the <> mean conditional infor-
mation <> of . Ygiven. X,then Eq.(3.8) tells us that the mutual information between. X
and . Yis the mean information of . Yminus the mean conditional information of . Y
given. X.

3.3 Entropic Concepts 35
Proposition 3.2 . D<>(<>p<>,<> q<>)<> ≥<> 0
Proof <> The following is a very simple inequality resulting from the graphic of loga-
rithm function:
. ln<> x<> ≤<> x<> −<> 1
therefore:
. ln
q
i
pi
≤
q
i
pi
−<> 1
multiplying by.piand summing over. ion both members:
.
∑
i
piln
q
i
pi
≤
∑
i
pi
(
q
i
pi
−<> 1
)
.
∑
i
piln
q
i
pi
≤
∑
i
qi−
∑
i
pi
because.
m
∑
i<> =<>1
pi=
m
∑
i<> =<>1
qi=<> 1(. pe. qare probability distributions):
.
∑
i
piln
q
i
pi
≤<> 0
or, by passing to logarithms in base. k:
.
∑
i
pilog
k
qi
pi
≤<> 0
then, reversing the logarithm fraction with the consequent change of sign, the opposite
inequality is obtained, that is:
.
∑
i
pilog
k
pi
qi
≥<> 0
D<>(<>p<>,<> q<>)<> ≥<> 0
□
Proposition 3.3 • .I<> (<>X<>,<> Y<> )<> =<> I<> (<>X<>,<> Y<> ) (symmetry)
• .I<> (<>X<>,<> Y<> )<> ≥<> 0 (non-negativity)
• . I<> (<>Y<>,<> X<>)<> =<> H<>(<>X<>)<> +<> H<>(<>Y<> )<> −<> H<>(<>X<>,<> Y<> )
Proof <> The ﬁrst equation follows directly from the deﬁnition of mutual informa-
tion, while the second one follows from non-negativity of Kullback-Leibler diver-
gence. The third equation follows from equations.I<> (<>X<>,<> Y<> )<> =<> H<>(<>X<>)<> −<> H<>(<>X<>|<>Y<> ) and

36 3 Information Theory
.I<> (<>Y<>,<> X<>)<> =<> H<>(<>Y<> )<> −<> H<>(<>Y<> |<>X<>) , by replacing in them the conditional entropies as they
result from.H<>(<>X<>|<>Y<> )<> =<> H<>(<>X<>)<> −<> H<>(<>X<>,<> Y<> ) and.H<>(<>Y<> |<>X<>)<> =<> H<>(<>Y<> )<> −<> H<>(<>Y<>,<> X<>) .<> □
Entropic divergence and mutual information provide two important properties of
entropy.
Proposition 3.4 <> The entropy of a random variable .Xreaches its maximum when. X
is uniformly distributed.
Proof <> If. nis the number of values assumed by.X(discrete and ﬁnite case), then:
. H<>(<>X<>)<> ≤<> log<> n<>.
In fact, if.Xis uniformly distributed we have:
. H<>(<>X<>)<> =−
n
∑
1
1
n
log
1
n
=−
n
n
log
1
n
=<> log<> n
If. udenotes the uniform probability distribution:
.
D<>(<>p<>,<> u<>)<> =
∑
x
p<>(<>x<>)<> log
p<>(<>x<> )
u<>(<>x<> )
=
∑
x
p<>(<>x<>)(log<> p<>(<>x<>)<> −<> log<> u<>(<>x<>) )
=
∑
x
p<>(<>x<>)<> log<> p<>(<>x<>)<> −
∑
x
p<>(<>x<>)<> log<> u (x)
=−<>H<>(<>X<>)<> +<> log<> n
therefore:
. −<> H<>(<>X<>)<> −<> log 1<>/<>n<> ≥<> 0
that is:
.H<>(<>X<>)<> ≤<> log<> n
□
Proposition 3.5
. H<>(<>X<>|<>Y<> )<> ≤<> H<>(<>X<>).
Proof
. I<> (<>X<>,<> Y<> )<> =<> H<>(<>X<>)<> −<> H<>(<>X<>|<>Y<> )
and:
. D<>(<>p<>,<> q<>)<> ≥<> 0
therefore:.I<> (<>X<>,<> Y<> )<> ≥<> 0 that is:

3.3 Entropic Concepts 37
. H<>(<>X<>)<> −<> H<>(<>X<>|<>Y<> )<> ≥<> 0
that gives the thesis.<> □
Let.H<>(<>X<>)be the uncertainty of. X, and.H<>(<>X<>|<>Y<> )be the conditional uncertainty of
.Xgiven. Y. Mutual information results to be equal to the decrease of the uncertainty
of.H<>(<>X<>)produced by.H<>(<>X<>|<>Y<> ):
Proposition 3.6
. I<> (<>X<>,<> Y<> )<> =<> H<>(<>X<>)<> −<> H<>(<>X<>|<>Y<> ).
Proof
. I<> (<>X<>,<> Y<> )<> =
∑
x<> ,<>y
p<>(<>x<>,<> y<>)<> log
p<>(<>x<>,<> y<>)
p<>(<>x<>)<>p<>(<>y<>)
but.p<>(<>x<>|<>y<>)<> =
p<>(<>x<> ,<>y<>)
p<>(<>y<>)
, then:
. I<> (<>X<>,<> Y<> )<> =
∑
x<> ,<>y
p<>(<>x<>,<> y<>)<> log
p<>(<>x<>|<>y<>)
p<>(<>x<>)
from which the following equations hold:
.
I<> (<>X<>,<> Y<> )<> =
∑
x<> ,<>y
p<>(<>x<>,<> y<>)<> log<> p (x<>|<>y)−
∑
x<> ,<>y
p<>(<>x<>,<> y<>)<> log<> p (x)
=−<>H(X<>|<>Y)−
∑
x<> ∈<> A
(
∑
y<>∈<>B
p(x<>,<> y)
)
log<> p<>(<>x<>)
=−<>H<>(<>X<>|<>Y<> )<> −
∑
x<> ∈<> A
p<>(<>x<>)<> log<> p<>(<>x<>)
=−<>H<>(<>X<>|<>Y<> )<> +<> H<>(<>X<>)
=<> H<>(<>X<>)<> −<> H<>(<>X<>|<>Y<> )
□
According to the previous proposition, mutual information .I<> (<>X<>,<> Y<> )can be seen
as the uncertainty of.Xreduced by the uncertainty of.Xwhen. Yis known. The term
.H<>(<>X<>|<>Y<> )is called <> equivocation<>, and corresponds to the uncertainty of.Xgiven. Y.
Mutual information allows us to give a mathematical formulation of the commu-
nication process between a sender and a receiver connected by means of a channel of
communication where noise is acting disturbing the communication. In this picture
sender and receiver are two information sources, or random variables.X<>,<> Y, respec-
tively, and a quantity, called <> channel capacity . Cmeasures the maximum amount of

38 3 Information Theory
information that can be transmitted through the channel in the time unity, where the
maximum is considered in the class of the probability distributions . passociated to
the sender. X:
. C<> =<> max
p
I<> (<>X<>,<> Y<> )
The second Shannon theorem will show that .Cresults to be equivalent to the max-
imum <> transmission rate<>, that is, the maximum quantity of information transmitted
by the sender, in the time unity, that can be correctly received by the receiver.
3.4 Codes
A <> code <> is a function from a ﬁnite set .Cof <> codewords <> or <> encodings<>, which are
strings over a ﬁnite alphabet, to a set .Dof data. In the following, we often will
identify a code with the set .Cof its codewords, by using the term <> encoding <> for
referring to the function from .Cto . D. When encoding is not injective, the code
is said to be <> redundant<>, because different codewords can encode the same datum.
When encoding is not surjective, the code is said to be <> degenerate<>. When encoding
is not deﬁnite for some codewords, the code is said to be <> partial. In the following
discussion, we will often assume 1–1 codes, with 1-to-1 encoding functions.
The <> genetic code <> is a code that is redundant and partial. It encodes 20 amino acids
with terns of symbols, called <> codons<>, over the alphabet .U<>,<> C<>,<> A<>,<> G . Many codons
may encode the same amino acid, and there are 3 of the 64 codons (UGA. UAA,
UAG) that do not encode any amino acid and are stop signals in the translation from
sequences of codons to sequences of amino acids (Table 3.1).
The Standard ASCII has only 7 bits (Table 3.2). Extended ASCII (ISO-Latin-
8859) has 8 bits.
In the ASCII code, a character is encoded by summing the column value with
the row value. For example: .N<> →<> 40<> +<> E<> =<> 4<>E . Then hexadecimal encodings are
transformed into binary values.
The <> digital information <> of datum . d, with respect to a code. C, is the length of the
string.α<> ∈<> Cencoding. d(the arithmetic mean of the codewords encoding. d,ifthey
are more than 1). Digital information is not a good measure of the information of
data. In fact, as we will see, given a code, another code can exist encoding the same
data with shorter codewords. This means that a measure of information, independent
from codes, should require the search for codewords of minimum lengths. One of
the main results of Shannon’s probabilistic approach is the ﬁrst Shannon theorem,
which explains as the entropy of an information source provides a minimal bound to
any digital measure of information.
A code .Cis <> preﬁx free<>,or <>instantaneous<>, if no codeword of .Cis a preﬁx of
another codeword of. C. When a code is preﬁx-free, it can be represented by a rooted
tree, called <> encoding tree<>, where codewords are the leaves of the tree. Starting from
the root, a number of edges equal to the symbols of the alphabet spring from the root,

3.4 Codes<> 39
Table 3.1 <> The amino-acids and the corresponding codons
Amino-acid
Name
LetterCodons
Arg Arginine
R. {<>CGU<>,<> CGC<>,<> CGA<>,<> CGG<>,<> AG A<>,<> AGG<>}
Leu Leucine
L. {<>UU A<>,<> UUG<>,<> CUU<>,<> CUC<>,<> CU A<>,<> CUG<>}
Ser Serine
S. {<>UCC<>,<> UCU<>,<> UCA<>,<> UCG<>,<> AGU<>,<> AGC<>}
Ala Alanine
A. {<>GCU<>,<> GCC,<> GC A<>,<> GCG<>}
Gly Glycine
G. {<>GGU<>,<> GGC<>,<> GGA<>,<> GGG<>}
Pro Proline
P. {<>CCU<>,<> CCC<>,<> CCA<>,<> CCG<>}
Thr Threonine
T. {<>ACU<>,<> ACC<>,<> AC A<>,<> ACG<>}
Va l Valine
V. {<>GUU<>,<> GUC<>,<> GU A<>,<> GUG<>}
Ile Isoleucine
I. {<>AUU<>,<> AUC<>,<> AU A<>}
Asn Aspargine
N. {<>AAU<>,<> AAC<>}
Asp Aspartate
D. {<>GAU<>,<> GAC<>}
Cys Cysteine
C. {<>UGU<>,<> UGC<>}
His Hystidine
H. {<>CAU<>,<> CAC<>}
Gln Glutamine
Q. {<>GAA<>,<> GAG<>}
Glu Glutamate
E. {<>CAA<>,<> CAG<>}
Lys Lysine
K. {<>AAA<>,<> AAG<>}
Phe Phenilananine
F. {<>UUU<>,<> UUC<>}
Tyr Tyrosine
Y. {<>UAU<>,<> UAC<>}
Met Methionine
M. {<>AU G<>}
Trp Tryptophan
W. {<>UGG<>}
Table 3.2 <> The Table of ASCII standard code
0
123456789ABCDEF
00 NULL
SOH STX ETX EOT ENQ ACK BEL BSTA B LFVTFFCRSOSI
10 DLE
DC1 DC2 DC3 DC4 NAK SYN ETB CAN EMSUB ESC FSGSRSUS
20 SPACE
!¨#$%&’()*+,–./
30 0
123456789:;.<=.>?
40 @
ABCDEFGHIJKLMNO
50 P
QRSTUVWXYZ[.\]ê_
60 ‘
abcdefghijklmno
70 p
qrstuvwxyz{|}˜DEL
each edge labelled by a symbol of the alphabet of. C. The same situation is repeated
for all the nodes of the tree that are not leaves. In such a way, each path from the root
to a leaf identiﬁes the sequence of the edges of the path, and the sequence of labels is a
codeword of. C. Therefore, by construction, codewords cannot be substrings of other
codewords. The encoding tree of Fig. 3.4 shows that that, passing from a node to
their sons, a partition of data is realized, and a leaf is obtained when a single datum is

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pohjoisrannikon säännöllisestä käyristymisestä johti hänet
päättämään, että senkautta oli osotettu joku fysikalisessa suhteessa
huomattava keskipiste. Kun hän etsi tätä suuremmoista keskipistettä,
huomasi hän, että Suuri pyramiidi oli juuri tässä keskipisteessä; tämä
sai hänet huudahtamaan: "Tällä muistomerkillä on fysikalisessa
suhteessa tärkeämpi asema kuin millään muulla ihmiskäsin tehdyllä
rakennuksella."
Pohjoisessa olevasta Sisäänkäytävästä vedetty viiva sattuu juuri
Egyptin rannikon pohjoisimpaan pisteeseen; ja ne viivat, jotka
vedetään rakennuksen koillis- ja luoteislävistäjille pitennyksiksi
sulkevat kummaltakin puolelta suistomaan väliinsä ja käsittävät niin
ollen tuon viuhkanmuotoisen Ala-Egyptinmaan. Ollen rakennettu
Gize-kallion pohjoisimmalle syrjälle ja näkyen sektorin tai viuhkan
muotoisen Ala-Egyptin yli, voi todellakin sanoa, että se on juuri sen
rajalla niinkuin myöskin sen nimellisessä keskipisteessä, kuten
profetta Jesaja sanoo. "Sinä päivänä on Herralla oleva alttari keskellä
Egyptinmaata, ja patsas (pyramiidi) on Herralla oleva sen rajalla. Ja
se on oleva merkiksi ja todistukseksi Herralle Sebaotille
Egyptinmaassa." — Toinen huomattava asianhaara on, että Suuri
pyramiidi sijaitsee maapallon maanpinnan maantieteellisessä
keskipisteessä — mukaan luettuina myöskin Pohjois- ja Etelä-
Amerikka, jotka olivat tuntemattomia vielä vuosisatoja pyramiidin
perustuksen laskemisen ja rakentamisen jälkeenkin.
SEN TIETEELLISET OPETUKSET.
Suuri pyramiidi ei puhu meille hieroglyfikirjotuksen eikä myöskään
piirustusten kautta vaan ainoastaan asemansa, rakenteensa ja
mittainsa kautta. Ainoat alkuperäiset merkit, jotka on huomattu

olevan, olivat "Kuninkaan huoneen" yläpuolella olevissa
"Konstruktioni-huoneissa"; pyramiidin huoneissa ja käytävissä
sitävastoin ei löydy mitään sellaisia. Suuren pyramiidin tieteelliset
opetukset jätämme mainitsematta säästääksemme tilaa, koska ei
edes yksi sadasta tavallisesta lukijasta ymmärtäisi tieteellisiä
termejä, niin että voisivat ymmärtää todistukset ja erittäinkin kun ne
eivät kuulu ollenkaan evankeliumiin, jota meidän tarkotuksemme on
esittää. Olkoon sentähden kylliksi, kun ainoastaan annamme
viittauksen siitä tavasta, jolla se opettaa tiedemiehiä. Esimerkiksi
huomataan perustuksen kohdalta neljän asemasivun pituudet
yhteenlaskettuna tekevän yhtä monta pyramiidi-kyynärää (noin 18
tuumaa) murto-osineen, kuin päiviä on neljässä vuodessa —
lukuunottaen karkausvuodenkin murto-osat. Lävistäjät koillisesta
lounaaseen ja luoteesta kaakkoon mitattuina perustuksesta tekevät
juuri yhtä monta tuumaa kuin on vuosia precessionalikaudessa.
Tämän jakson olivat tähtitieteilijät jo otaksuneet 25,827 vuodeksi ja
Suuri pyramiidi vahvistaa heidän päätelmänsä. Etäisyyden aurinkoon,
arvellaan, ilmaisee Suuri pyramiidi korkeutensa ja kulmansa kautta
91,840,270 engl. penik. (147,770,994 km), mikä vastaa melkein
täydellisesti niitä viimeisiä numeroita, joihin tähtitieteilijät ovat
tulleet. Tähtitieteilijät olivat äskeisiin aikoihin saakka laskeneet
tämän etäisyyden 90—96 miljoonaksi engl. penik. ja heidän
viimeinen laskelmansa ja lopputuloksensa on 92 miljoonaa engl.
penik. (148,028,000 km). Suurella pyramiidilla on myöskin oma
tapansa ilmaista mitä oikeimmat ohjeet kaikille mitoille ja painoille,
mitkä perustuvat maan suuruuteen ja painoon, jotka sen myöskin
sanotaan ilmaisevan.
Teol. t:ri Josef Seiss huomauttaa, kertoessaan tämän
majesteetillisen todistajan tieteellisistä todistuksista ja sen asemasta,
seuraavaa:

"Tämä ihmeellinen rakennus sisältää vielä suuremmoisemmankin
ajatuksen. Sen viidestä kulmasta on erittäinkin silmäänpistävä yksi,
jossa sen (pyramiidin) sivut ja kaikki rajaviivat päättyvät. Se on
huippukulmakivi, joka nostaa puolipäivän aikaan juhlallisen
sormensa aurinkoa kohti ja ilmaisee etäisyytensä kautta
perustuksesta keskimääräisen etäisyyden maasta aurinkoon. Ja jos
me menemme takaisin siihen vuoteen, minkä pyramiidi itse ilmaisee
ja katsomme mitä tämä sormi silloin näytti puoliyönä, niin saamme
vielä paljon ylevämmän selityksen. Tiede on viimeksi huomannut,
että aurinko ei ole mikään kuollut keskipiste, jonka ympäri kiertää
kiertotähtiä, mutta joka itse olisi paikallaan oleva. Nykyään on tullut
vahvistetuksi, että myöskin aurinko liikkuu, vieden mukanaan
loistavan pyrstötähtiseurueensa noiden ja omien saattolaistensa
kanssa jonkun tavattoman paljon suuremman keskipisteen ympäri.
Tähtitieteilijät eivät ole vielä täydellisesti yksimielisiä siitä, mikä tai
missä tämä keskipiste on. Muutamat kumminkin arvelevat
löytäneensä sen suunnan olevan Plejadit (Seulaset) ja erittäinkin
niissä tähti Alcyone, keskimäinen noista kuuluisista plejadi-tähdistä.
Kunnia tämän keksimisestä kuuluu etevälle saksalaiselle
tähtitieteilijälle prof. J. Maedler'ille. Tähti Alcyone on siis, niin paljon
kuin tiede kykenee päättämään, se keskiyön valtaistuin, missä on
koko painolaki-järjestelmän keskipiste ja mistä Kaikkivaltias hallitsee
maailman kaikkeutta. Ja tässä on huomattava se tosiasiain
ihmeellinen vastaavaisuus, että Suuren pyramiidin rakennusvuonna
syyspäiväntasauksen keskiyönä ja siis vuoden oikeana
alkamispäivänä [Juutalaisen vuoden alku, joka alkaa sovintopäivänä,
kuten osotetaan Raamatun tutkistelujen II osassa] jollaisena se
vieläkin säilyy monien kansanheimojen perinnäiskertomuksissa,
olivat Seulaset jakaantuneet tämän pyramiidin puolipäiväpiirin
kohdalla niin että tähti Alcyone (Tauri) oli juuri tällä linjalla. Tässä on

siis laatuaan mitä korkein ja ylevin viittaus, minkä inhimillinen tiede
koskaan on ainoastaan kyennyt ilmaisemaan ja mikä näyttää luovan
ennen aavistamattoman valtavan merkityksen Jumalan puheeseen
Jobille, kun hän sanoi: 'Taidatko solmea Plejadien suloisen
vaikutuksen'" (engl. k.)?
SEN TODISTUS LUNASTUSSUUNNITELMAAN NÄHDEN.
Samalla kuin jokainen piirre siitä mitä Suuri pyramiidi opettaa on
tärkeä ja mielenkiintoa herättävä, niin on sen äänetön, mutta hyvin
puhuva vertauskuvallisuus Jumalan suunnitelmasta —
aikakausiensuunnitelmasta se, mikä eniten kiinnittää meidän
mieltämme. Olisi kumminkin mahdotonta ymmärtää Jumalan
suunnitelmaa Suuren pyramiidin esittämässä muodossa, jollemme
ensin olisi löytäneet sitä suunnitelmaa Raamatusta. Mutta sittekun
me olemme nähneet sen esitettynä siellä, on hyvin uskoa
vahvistavaa nähdä se taas täällä niin kauniisti ääripiirteettäin
esitettynä; ja edelleen on huomattavaa että sama suuri Luoja antaa
tunnustuksensa ja todistuksensa sekä luonnon totuuksille että
ilmestyksen totuuksille tässä ihmeellisessä kivi-"todistuksessa".
Tällä alalla on, ulkoapäin katsottuna, Suuren pyramiidin
opetuksella kaunis merkitys, sen kautta että se esittää Jumalan
suunnitelman täytäntöön vietynä, sellaisena kuin se tulee olemaan
tuhatvuotiskauden lopussa. Kruununa siinä tulee olemaan Kristus,
tuo tunnustettu pää yli kaikkien; ja jokainen muu kivi tulee ihan
täydellisenä olemaan, sopivasti liitetty tuohon ihanaan
rakennukseen. Kaikki karsiminen, puhdistaminen ja sovitteleminen
tulee silloin olemaan loppuun saatettu ja kaikki tulevat riippumaan ja
olemaan sidottuina toisiinsa ja päähänsä rakkaudella. Jos Suuri

pyramiidi esittää Jumalan suunnitelman kokonaisuudessaan,
täydellisenä, täytyy sen huippukulmakiven esittää Kristusta, jonka
Jumala on korottanut korkealle, olemaan kaikkien päänä. Ja että se
todellakin esittää Kristusta, käy selville, ei ainoastaan sen
täydellisestä sopivaisuudesta vertauskuvaksi Kristuksesta [Katso I
osa, 5 luku; myös aikakausien kartta, I osa x, y, z, W.] vaan myöskin
niistä lukuisista profettain, apostolien ja meidän Herramme
Jeesuksen itsensäkin tekemistä viittauksista tähän vertauskuvaan.
Jesaja (28: 16) viittaa Kristukseen "kalliina kulmakivenä". Sakarja
(4: 7) tarkottaa sen asettamista tuon valmistetun rakennuksen
huipulle suuren riemun vallitessa sanoessaan: "Hän tuo esille
harjakiven, riemuhuudon kaikuessa: armo, armo olkoon hänelle!"
Epäilemättä oli silloin kun Suuren pyramiidin pää eli huippukivi
laskettiin suuri riemu rakennusmiesten ja kaikkien niiden kesken,
joiden mieltä kiinnitti saada nähdä, kun kruunu asetettiin
loppuunsaatetun rakennuksen päälle. Job (38: 6, 1) puhuu myöskin
siitä ilosta, joka vallitsi silloin kun tuo ylin kulmakivi laskettiin, ja hän
ilmaisee erittäin huippukulmakiven eli sen kulmakiven, joka kruunasi
kaiken, mainitsemalla ensin nuo toiset neljä kulmakiveä, sanoessaan:
"Mihinkä ovat sen pohjaperustukset upotetut, tahi kuka on sen
kulmakiven laskenut, kun aamutähdet ynnä iloitsivat ja kaikki
Jumalan pojat riemuitsivat?" Profetta Daavid tarkottaa myöskin
meidän Herraamme ja käyttää kuvakieltä, joka hyvin vastaa sitä mitä
tämä Egyptin "todistaja" käyttää. Hän sanoo profetallisesti
tulevaisuuden näkökannalta: "Siitä kivestä, jonka rakentajat
hylkäsivät on tullut huippukulmakivi. Jehovan työtä on tämä. Se on
ihmeellistä meidän silmissämme. Tämä on se päivä
(tuhatvuotispäivä, Kristuksen ihanuuden päivä maailman päänä ja
hallitsijana), jonka Jehova teki; me tahdomme iloita ja riemuita
siitä". (Ps. 118: 22—24 engl. k.). Lihallinen Israel ei osannut ottaa

Kristusta vastaan huippukivenään ja tuli sentähden hyljätyksi
olemasta Jumalan erityisenä huoneena — hengellinen Israel
rakennettiin sen sijaan Kristuksessa, Päässä. Ja me muistamme että
Herramme sovitti juuri tämän ennustuksen itseensä ja osotti, että
hän oli hyljätty kulmakivi, ja että Israel, rakennusmiestensä, pappien
ja fariseusten kautta, olivat ne, jotka hylkäävät hänet. (Matt. 21: 42,
44; Apt. 4: 11).
Kuinka täydellisesti Suuren pyramiidin huippukivi valaiseekaan
kaikkea tätä! Huippukivi, joka valmistettiin ensin, oli työntekijöille
esikuvana ja mallina koko rakennuksesta, jonka kaikki kulmat ja
mittasuhteet tuli sovittaa sen mukaan. Mutta me voimme helposti
kuvitella itsellemme, että ennenkuin tämä huippukivi hyväksyttiin
esikuvaksi koko rakennukselle, hylkäsivät rakennusmiehet sen
arvottomana, koska heidän joukossaan muutamat eivät voineet
ajatella sille mitään sopivaa paikkaa: sillä sen viisi sivua, viisi nurkkaa
ja kuusitoista kulmaa tekivät sen sopimattomaksi rakennukseen,
kunnes tarvittiin juuri huippukiveä, ja silloin ei taas kelvannutkaan
mikään muu kivi. Kaikkina vuosina, jolloin rakentaminen jatkui, oli
tuo ylin kulmakivi "loukkauskivi" ja "pahennuksen kallio", niille, jotka
eivät tunteneet sen käytäntöä ja paikkaa; aivan niinkuin Kristus on ja
tulee yhä edelleen olemaan monelle, kunnes ovat nähneet hänet
korotettuna Jumalan suunnitelman Pää-kulmakiveksi.
Pyramiidin muoto kuvaa täydellisyyttä ja puhuu meille
vertauskuvin Jumalan suunnitelmasta, osottaen, että "hän aikain
täyttyessä on kokoava yhteen (sopusointuiseksi perheeksi, vaikka eri
olemassaolotasoilla) kaikki taivaissa ja maan päällä, Pään, Kristuksen
alle" — ja että kaikki, jotka eivät tule yhdenmukaisiksi, karsitaan pois
— Ef. 1: 10; 2: 20, 22. — Wilson' Diaglott (kreikk. engl. U.T.).

KUINKA SUUREN PYRAMIIDIN SISÄINEN RAKENNE
ILMAISEE PELASTUSSUUNNITELMAN ÄÄRIVIIVAT.
Mutta kuten tuon suuren rakennuksen ulkopuolinen todistus siten on
täydellinen ja sopusoinnussa Jumalan kirjotetun ilmestyksen kanssa,
samaten on sen sisäinen rakenne vielä ihmeellisempi. Kun sen
ulkonainen muoto valaisee täytäntöön saatetun Jumalan
pelastussuunnitelman tulosta [Aikakausien kartta I osassa], niin
ilmaisee ja valaisee sen sisäinen rakenne taas jokaisen tämän
suunnitelman huomattavamman piirteen sellaisena kuin se on
kehittynyt aikakaudesta aikakauteen aina ihanaan ja täydelliseen
täytäntöönsä asti. Tässä esittävät kivet eri korkeuksilla eli tasoilla
kaikkien niiden täydellisyyttä, jotka Kristuksen meidän Päämme
alaisina tehdään yhdenmukaisiksi Jumalan täydellisen tahdon
kanssa, kuten me jo olemme nähneet Raamatun todistuksesta.
Toiset tulevat täydellisiksi inhimillisellä tasolla, toiset henkisellä ja
toiset taas jumalallisella tasolla eli luonnossa. Siten on "Kuninkaan
huoneen" lattia viidennelläkymmenennellä muurauskerroksella,
"Kuningattaren huoneen" kahdennellakymmenellä viidennellä
kerroksella, ja "Ensimäisen nousevan käytävän" alin pää tulisi, jos
sitä jatkettaisi "Tulpan" läpi, olemaan pyramiidin asemaviivalla, kuten
heti tullaan näyttämään. Niin ollen näyttää Suuri pyramiidi
asemaviivastaan ylöspäin olevan tunnusmerkkinä Jumalan koko
ihmiskuntaa varten valmistamasta pelastussuunnitelmasta eli
synnistä ja kuolemasta nostamisesta. Asemaviiva vastaa niinollen
sitä vuotta, jolloin Jumala vahvisti lupauksensa esikuvalliselle
Israelille — kohottamisen eli pelastamisen alkua.
Kehotetaan tutkimaan huolellisesti piirrosta, joka esittää tuon
ihmeellisen rakennuksen sisäisen rakenteen. Suurella pyramiidilla on
ainoastaan yksi ainoa oikea "Sisäänkäyntikäytävä". Tämä käytävä on

säännöllinen, mutta matala ja alaspäin viettävä ja johtaa alas
pieneen huoneeseen eli "Maanalaiseen huoneeseen", joka on
hakattu kallioon. Tämä huone on omituinen rakennustavaltaan sen
kautta että katto on hyvin puhdistettu jota vastoin sivuja on
ainoastaan alotettu ja pohja on aivan ryhmyinen ja siistimätön.
Tämä on antanut aihetta ajatella "pohjatonta kuilua"
("pimeydenkuilua", uusi Raamatun käännös), jota kuvausta
Raamatussa käytetään ilmaisemaan onnettomuutta, unholaa ja
tyhjäksitekemistä. "Sisäänkäyntikäytävä" esittää sopivalla tavalla
ihmissuvun alaspäinmenevää kulkua perikatoa kohti; "Maanalainen
huone" taas valaisee eriskummallisen rakennustapansa kautta sitä
suurta hädän aikaa, onnettomuutta, perikatoa, synnin palkkaa, mihin
alaspäin vievä kulku johtaa.
"Ensimäinen nouseva käytävä" on jotakuinkin yhtä iso kuin
"Sisäänkäyntikäytäväkin", mistä se saa alkunsa. Se on pieni, matala
ja vaikeasti noustava, mutta se avautuu ylipäässään suureen,
muhkeaan käytävään, jota kutsutaan "Suureksi galleriaksi", jonka
katto on seitsemän kertaa korkeampi kuin niiden käytävien, jotka
johtavat sinne. Matalan "Nousevan käytävän" arvellaan esittävän
lakitaloutta ja Israelia kansakuntana Egyptistä lähdöstä alkain. Siinä
jättivät he maailman kansakunnat ja heidän alaspäin vievän tiensä
ollakseen Jumalan pyhänä kansana ja hänen lakinsa alaisena —
aikoen senjälkeen vaeltaa ylöspäin vievää ja vaikeampaa tietä kuin
pakana-maailma, s.o. aikoen pitää lain. "Suuren gallerian" katsotaan
esittävän evankelisen kutsun aikaa — yhä edelleen ylösnouseva ja
vaikea, mutta ei matala ja ahdas niinkuin sen takana oleva. Tämän
käytävän suurempi korkeus ja avaruus kuvaa hyvin niitä
suuremmoisia toiveita ja sitä suurempaa vapautta, joka kuuluu
kristilliseen armotalouteen.

"Suuren gallerian" lattian alapään tasalta alkaa "Vaakasuora
käytävä", joka on sen (Suur. gall.) alla ja joka johtaa pieneen
huoneeseen, jota tavallisesti kutsutaan "Kuningattaren huoneeksi".
"Suuren gallerian" ylipäässä on toinen matala käytävä, joka johtaa
pieneen huoneeseen, jota kutsutaan "Esihuoneeksi", jonka rakenne
on hyvin omituinen ja joka muutamissa herättää ajatuksen koulusta
— paikasta, missä opetetaan ja koetellaan.
Mutta huomattavin huone Suuressa pyramiidissa sekä suuruuteen
että aseman tärkeyteen nähden on vähän etäämpänä oleva toisen
matalan käytävän kautta "Esihuoneesta" erotettu huone. Tämä on
tunnettu "Kuninkaan huoneen" nimellä. Sen päällä on muutamia
pieniä huoneita, joita kutsutaan "Konstruktionihuoneiksi". Mitä ne
merkitsevät, jos niillä joku merkitys on, niin ei niillä ainakaan ole
ihmisiin tahi muihinkaan jaloilla käveleviin olentoihin nähden mitään
merkitystä, vaan henkiolentoihin nähden; sillä huomattakoon, että
vaikka sivut ja katto ovat suorakulmaisia ja puhdistettuja, niin ei
niissä yhdessäkään ole lattiata. "Kuninkaan huoneessa" on "Arkku"
eli kivilaatikko; ainoa huonekalu, mikä on Suuressa pyramiidissa.
"Kuninkaan huoneen" ilmanvaihdosta on huolehdittu kahden
ilmaputken kautta, jotka lävistävät sen seinät vastakkaisilta puolilta
ulottuen aina ulkopintaan asti — jotka rakentajat ovat jättäneet tätä
tarkotusta varten. Muutamat ovat lausuneet sen arvelun että löytyy
vielä toisia huoneita ja käytäviä tulevaisuudessa löydettäviksi, mutta
me emme ole samaa mielipidettä: meistä tuntuu että ne käytävät ja
huoneet, jotka jo on löydetty täydelleen ilmaisevat Jumalan aikoman
tarkotuksen todistaa jumalallinen suunnitelma kokonaisuudessaan.
"Suuren gallerian" ala- eli pohjoispään länsisivulta suuntautuu
alaspäin säännötön käytävä, jota kutsutaan "Kaivoksi" ja joka johtaa
alaspäin vievään "Sisäänkäyntikäytävään". Se kulkee luonnolliseen

kallioon hakatun "Luolan" läpi, yhteys tämän käytävän ja "Suuren
gallerian" välillä on aivan järjestämättömässä kunnossa. Näyttää siltä
että käytävä "Kuningattaren huoneeseen" alunperin on ollut
peitossa, senkautta että "Suuren gallerian" lattialaatat peittivät sen;
ja että myöskin "Kaivon" suu on ollut kivilaatan peittämä. Mutta nyt
on "Suuren gallerian" alapään suu revitty pois, ja tämän kautta on
käytävä "Kuningattaren huoneeseen" avautunut ja "Kaivo" jäänyt
avoimeksi. Ne jotka ovat olleet siellä ja tutkineet asiaa, sanovat, että
näyttää siltä kuin "Kaivon" suussa olisi tapahtunut räjähdys, joka on
rikkonut sen alhaaltapäin. Meidän mielipiteemme kumminkin on, että
ei mitään sellaista räjähdystä koskaan ole tapahtunut; vaan että
rakennusmiehet ovat jättäneet asianhaarat sellaisiksi kuin ne ovat,
tarkotuksella saada ilmaistuksi samaa kuin olisi saatu ilmaistuksi
tuon luulotellun räjähdyksen kautta, jota kosketellaan tuonnempana.
Sellaisena kuin asianhaarat nyt ovat, ei yhtään näistä kivistä ole
löydettävissä, ja olisi ollut hyvin vaikeata poistaa ne.
"Suuren gallerian" yli eli eteläpäähän ulottuu "Esihuoneen" ja
"Kuninkaan huoneen" lattia muodostaen "Suuren gallerian" ylipäähän
jyrkän kaiteen eli korkean portaan. Tämä porras (askel) on
eteläseinästä kuudenkymmenen yhden tuuman (n. 1,55 m) päässä.
"Suuren gallerian" eteläpään seinä on myös yhdessä suhteessa
omituinen: se ei ole kohtisuora, vaan kallistuu pohjoiseen päin —
seitsemän tuumaa (n. 18 cm) katonrajassa — ja juuri ylimmässä
kohdassa on aukko eli käytävä, joka on "Kuninkaan huoneen"
yläpuolella olevien "konstruktioni-huoneiden" yhteydessä.
Pyramiidin käytävät ja lattiat ovat kalkkikiveä, ylimalkaan koko
rakennuskin, lukuunottamatta "Kuninkaan huonetta", "Esihuonetta"
ja niiden välistä käytävää, missä lattiat ja katot ovat graniittia. Ainoa
pala graniittia muualla koko rakennuksessa on graniitti-"Tulppa", joka

on lujasti tilkitty "Ensimäisen nousevan käytävän" alapäähän.
Sellaisena kuin rakentajat alunperin jättivät "Ensimäisen nousevan
käytävän" oli se alapäästään, mistä se on yhteydessä
"Sisäänkäyntikäytävän" kanssa, suljettu erittäin sopivalla
kulmikkaalla kivellä; ja tämä oli tehty niin hyvin, että "Ensimäinen
nouseva käytävä" oli tuntematon, kunnes silloin kun "oikea aika" oli
tullut kivi putosi. Hyvin likellä "Ensimäisen nousevan käytävän"
alapäätä ja heti sulkukiven takana, oli graniitti-"Tulppa", joka on
tehty kiilanmuotoiseksi ja on nähtävästi aijottu pysymään siinä,
koska se toistaiseksi on vastustanut kaikkia poistamisyrityksiä. Vaikka
vanhat tunsivat hyvin "Sisäänkäyntikäytävän", minkä
historiankirjottajat vahvistavat, oli kuitenkin arabialainen kalifi Al
Mamoun nähtävästi tietämätön sen täsmällisestä asemasta, paitsi
että kansantarinat sijottivat sen pyramiidin pohjoissivulle, kun hän v.
825 j.K. suurilla kustannuksilla mursi sisäänkäyntikäytävän toivoen
löytävänsä ihmeellisiä aarteita. Mutta vaikka se sisältääkin laajoja
tietopuolisia aarteita, joita nyt pidetään arvossa, niin ei se sisältänyt
mitään sitä lajia aarteita, joita arabialaiset etsivät. Heidän
vaivannäkönsä ei ollut kumminkaan aivan turha; sillä heidän
työskentelynsä aikana vieri se kivi, joka sulki nousevan käytävän,
pois paikaltaan ja putosi "Sisäänkäyntikäytävään" ilmaisten
salaisuuden, paljastamalla "Ensimäisen nousevan käytävän".
Arabialaiset luulivat että he vihdoinkin olivat löytäneet tien kätketyille
rikkauksille, ja kun eivät kyenneet poistamaan graniitti-"Tulppaa",
mursivat he käytävän sen viereen, koska oli paljon helpompi hakata
pehmeämpää kalkkikiveä.
SUUREN PYRAMIIDIN AIKAKAUSIEN SUUNNITELMAA
KOSKEVA TODISTUS.

Eräässä kirjeessään prof. Smyth'ille sanoi tuo nuori skotlantilainen,
hra Robert Menzies, joka ensin esitti arvelun, että sillä, mitä Suuri
pyramiidi opettaa on myöskin uskonnollinen eli Messias puolensa,
seuraavaa: —
Suuren gallerian pohjoispään alusta nousevassa asteikossa alkavat
meidän Vapahtajamme elämänvuodet ilmaistuina niin, että tuuma
vastaa vuotta. Kolmekymmentäkolme tuuma-vuotta vie meidät siis
keskelle "Kaivon" suuta.
"Kaivo" on niin sanoaksemme, tämän kaiken avain. Se esittää sekä
meidän Herramme kuolemaa ja hautausta että myöskin hänen
ylösnousemistaan. Tämä jälkimäinen osotetaan sen jo huomautetun
piirteen kautta, että "kaivon" suu ja ympäristö näyttää siltä kuin
räjähdys olisi alhaaltapäin särkenyt sen. Niin särki meidän Herramme
kuoleman siteet ja toi senkautta elämän ja kuolemattomuuden
valoon — avasi uuden tien elämään. (Hebr. 10: 20). Murtuneet
lohkareet, jotka ympäröivät tämän "Kaivon" ylintä osaa näyttävät
sanovan, ettei ollut mahdollista, että kuolema olisi voinut pitää
hänet. (Apt. 2: 24.) Kuten "Kaivo" oli ainoa tie jokaiseen noista
Suureen pyramiidin nousevista käytävistä, niin on meidän
Lunastajamme kuolema ja ylösnousemus myöskin ainoa
mahdollisuus langenneen suvun jäsenille saavuttaa joku elämäntaso.
Kuten "Ensimäinen nouseva käytävä" oli siellä, mutta oli
luoksepääsemätön, niin oli myöskin juutalainen eli lakiliitto elämän
tie tai tarjous, mutta käyttämätön ja saavuttamaton elämän tie: ei
kukaan langenneen suvun jäsen ole koskaan voinut saavuttaa tai ole
koskaan saavuttanut elämää kulkemalla sen määräämää suuntaa. "Ei
yksikään ihminen tule vanhurskautetuksi lain teoista" elämään.
(Room. 3: 20.) "Kaivon" vertauskuvaama lunastus on ainoa tie, jonka
kautta kukaan tuomitun suvun jäsenistä voi saavuttaa tuon

suuremmoisen hyvän — joka on valmistettu jumalallisen
suunnitelman kautta — ijankaikkisen elämän.
Vuosia ennen sen arvelun esittämistä, että "Suuri galleria"
esikuvaa kristillistä armotaloutta, oli prof. Smyth tähtitieteellisten
havaintojen kautta vahvistanut vuosiluvun pyramiidin rakentamiselle
v. 2170 e.K. Ja kun herra Menzies esitti että jokaisen "Suuren
gallerian" lattialinjan tuuma esikuvaa yhtä vuotta, johtui jonkun
mieleen, että jos tämä teoria olisi oikea, pitäisi, jos lattialinja
mitattaisiin takaperin "Suuren gallerian" alareunasta alkain alas
pitkin "Ensimäistä nousevaa käytävää" siihen asti missä se yhtyy
"Sisäänkäyntikäytävään" ja siitä taas ylöspäin pitkin
"Sisäänkäyntikäytävää" pyramiidin sisäänkäytävään, tulisi tämän
osottaa tai ilmaista joku vastaavaisuus ja niin todistaa pyramiidin
rakennusvuosi kuten myöskin tuuma-vuosi teoria oikeaksi. Tämä,
vaikka ei ollutkaan mahdotonta oli vaikea koe, ja eräs siviili-insinööri
lähetettiin käymään pyramiidissa ja uudestaan ottamaan hyvin tarkat
mitat sen käytävistä, huoneista j.n.e. Tämä oli v. 1872; ja hänen
tiedonantonsa olivat äärimmäiseen asti vahvistavia. Hänen
mittauksensa osottavat äsken kuvatun lattialinjan olevan 2170 1/2
tuumaa "sisäänkäyntikäytävän" seinissä olevaan hyvin hienosti
viivattuun viivaan asti. Niin muodoin on sen rakennusvuosi tullut
kahdenkertaisesti vahvistetuksi; ja sen ohessa osotetaan että
lattialinjoilla sen käytävissä on sekä historiallinen että aikalaskullinen
merkitys, mitkä vielä yleisesti tulevat olemaan "todistukseksi Herralle
Egyptin maassa".
Tässä, kiitos olkoon niiden tarkkojen käytävistä otettujen mittojen,
jotka prof. Smyth on tehnyt, saatamme tulla siihen mikä on meille
verrattomasti mieltäkiinnittävintä niistä piirteistä, joita "todistaja"
vielä todistaa.

Kun me ensin näimme mitä jo on sanottu suuren pyramiidin
todistuksista, sanoimme heti: Jos tämä todellakin on Raamattu
kivessä; jos se on asiakirja tuon suuren maailmankaikkeuden
Rakennusmestarin salaisista suunnitelmista; asiakirja, joka ilmaisee
hänen edeltäpäin tietämisensä ja viisautensa, täytyy ja tulee sen olla
täydessä sopusoinnussa hänen kirjotetun Sanansa kanssa. Se seikka,
että pyramiidin salaisuudet ovat olleet kätketyt maailmanhistorian
kuudennen vuosituhannen loppuun asti, mutta että se nyt alkaa
ilmaista todistuksiansa sen mukaan kuin tuhatvuotispäivä edistyy, on
täydellisesti sopusoinnussa Raamatun kanssa, jonka runsaat
todistukset Jumalan ihmeellisestä suunnitelmasta ovat myöskin olleet
salatut maailman perustamisesta asti ja nyt vasta alkavat loistaa
täydellisyydessään ja ihanuudessaan.
Me olemme jo esittäneet edellisissä osissa ja tämän osan
edellisissä luvuissa Raamatun selvät todistukset, jotka ilmottavat että
me olemme uudenajan kynnyksellä — että tuhatvuotispäivä koittaa
ja että maan hallitus vaihtuu "tämän maailman ruhtinaalta" ja hänen
uskollisiltaan hänelle "jolle se kuuluu" (oston kautta) ja hänen
uskollisille pyhilleen. Me olemme nähneet, että vaikka tämän
vaihdoksen tulos tulee koitumaan suureksi siunaukseksi, niin tulee
kuitenkin vaihdosaika, jolloin nykyinen ruhtinas, tuo "väkevä",
sidotaan ja hänen ja hänen huonekuntansa ajetaan pois vallasta
(Matt. 12: 29; Ilm. 20: 2), olemaan vaikea hädän aika.
Raamatullinen ajanlasku, jota olemme tarkastaneet osottaa, että
hädän-ajan alku oli käsillä Kristuksen toisessa tulemisessa
(lokakuussa 1874), jolloin kansojen tuomitseminen tulisi alkamaan
Herran päivän valaisevien vaikutusten vallitessa. Tämä osotetaan
Suuressa pyramiidissa seuraavasti: —

"Alaslaskeva käytävä", joka johtaa suuren pyramiidin
sisäänkäytävästä "Kuiluun" eli "Maanalaiseen huoneeseen" kuvaa
maailman kulkua ylimalkaan (tämän maailman ruhtinaan hallitessa)
suureen hädän päivään ("kuiluun") asti, missä paha saatetaan
loppuunsa. Tämän aikakauden mittaaminen ja ajan määrääminen,
jolloin hädän luola saavutetaan on helppoa, jos meillä on joku
aikamäärä — piste pyramiidissa, mistä me lähdemme. Meillä on
tämä merkki "Ensimäisen nousevan käytävän" ja "Suuren gallerian"
yhtymäkohdassa. Tämä piste ilmaisee meidän Herramme Jeesuksen
syntymän, kuten "Kaivo" 83 tuumaa kauempana ilmottaa hänen
kuolemaansa. Siis, jos me mittaamme takaperin pitkin "Ensimäistä
nousevaa käytävää" siihen asti, missä se yhtyy
"Sisäänkäyntikäytävään", on meillä huomattavana joku määrätty
päivämäärä alaspäin johtavalla käytävällä. Tämä mitta on 1542
tuumaa (n. 39 m) ja ilmaisee siis v. 1542 e.K. aikamääräksi tässä
pisteessä. Mittaamalla sitte alaspäin pitkin "Sisäänkäyntikäytävää"
tästä pisteestä "Kuilun" sisäänkäytävään, joka "Kuilu" esikuvaa sitä
suurta hädän aikaa ja hävitystä, jolla tämä aikakausi päättyy, kun
paha syöstään pois vallasta, huomaamme sen olevan 3457 tuumaa
(n. 86,5 m), jotka vertauskuvaavat 3457 vuotta yllä mainitusta ajasta
alkain 1542 e.K.; tämä laskelma osottaa 1915 ilmettävän hädän ajan
alun; sillä 1542 vuotta e.K. ja 1915 vuotta j.K. tekevät yhteensä
3457 vuotta. Niin muodoin todistaa pyramiidi että 1914 loppu oli
aikalaskullisessa suhteessa hädän-ajan alku, hädän-ajan, jonka
vertaista ei koskaan ole ollut, sitte kun ihmisiä rupesi olemaan —
eikä myöskään koskaan tule. Ja niin muodoin osottautuu, että tämä
"todistaja" vahvistaa täydelleen Raamatun todistukset tämän asian
suhteen, kuten osotetaan RAAMATUN-TUTKISTELUJEN II OSAN
7:ssä luvussa "Rinnakkaisia armotalouksia".

Ei kenenkään pidä epäillä, että nuo neljäkymmentä tuomion ja
vaikeuden vuotta alkoivat syksyllä 1874, sentähden että hätä ei ole
saavuttanut niin hirveää ja sietämätöntä tilaa; ja että muutamissa
suhteissa, tuon vuoden jälkeen tämä "elonkorjuu"-aika on ollut suuri
tiedon edistymisen kausi. Muista, että kaikki tämä osotetaan
Suuressa pyramiidissa ja että sen tekee selväksi piirustus "Kuilusta",
jonka prof. Smyth piirusti ilman minkäännäköistä sovittelun
aikomusta tähän nähden.
Sitäpaitsi tulee meidän muistaa, että Raamattu selvästi osottaa,
että hädän ajan tuomiot alkavat nimiseurakunnasta, valmistuksena
sen kukistumiselle ja itsekkäästä taistelusta työn ja kapitaalin välillä,
mitkä molemmat nyt järjestäytyvät lähestyväksi hädän päiväksi.
Tämän alimman huoneen eli "Kuilun" muoto ja viimeistely ovat
omituisen merkitseviä. Katto ja osittain sivutkin ovat säännöllisiä,
mutta mitään lattiata ei ole — sen ryhmyinen, keskeneräiseltä
näyttävä pohja laskee itää kohden aina enemmän ja enemmän,
antaen aihetta nimeen "Pohjaton kuilu", joksi sitä joskus kutsutaan.
Tämä huone puhuu vapaudesta ja rajottamattomuudesta kuten
myöskin vaikeuksista, korotuksesta kuten myöskin alennuksesta; sillä
kun kulkija saapuu sinne kankeana ja väsyneenä siitä kyykistyneestä
asennosta, mihin "Sisäänkäyntikäytävän" ahtaus on hänet
pakottanut, huomaa hän täällä että ei ainoastaan alaspäin ole
suurempi syvyys, hyvin epätasaisella ja murtuneella "hädän lattialla",
vaan hän huomaa myöskin ylhäälläpäin suuren korotuksen, sen
kautta että osa tämän huoneen katosta on hyvin paljon
korkeammalla kuin sinne johtava käytävä, mikä merkitsee
melkoisesti laajentunutta tilaa hänen ymmärryselimillensä.

Kuinka hyvin tämä vastaakaan todellisuutta. Emmekö jo voi nähdä
että vapaudenhenki on saavuttanut sivistyneiden kansakuntien
kansanjoukot. Me emme pysähdy katsomaan johdonmukaisuutta ja
epäjohdonmukaisuutta niihin vapauksiin nähden, joita joukot
tuntevat ja joita he vaativat, vaikka tämäkin on ilmaistu tässä
huoneessa katon korkeuden ja pohjan laskemisen kautta: me
huomautamme ainoastaan, että valo meidän päivänämme — Herran
päivänä — synnyttää vapauden hengen; ja kun vapauden henki
tulee kosketukseen ylpeyden, rikkauden ja vallan kanssa niihin
nähden, joilla vielä on valtaa, tulee se aiheuttamaan hädän-ajan,
jonka Raamattu vakuuttaa lopulta tulevan hyvin suureksi. Vaikka se
vielä tuskin on alkanut, näkevät kuninkaat ja keisarit, suurmiehet ja
kapitalistit ja kaikki sen tulevan ja "ihmiset menehtyvät peljätessään
ja odottaessaan sitä, mikä kohtaa maanpiiriä"; sillä taivaan voimat
ovat järkkymässä ja kukistetaan ne lopulta. Nykyisen pahan
maailman väärät — valtiolliset, yhteiskunnalliset ja uskonnolliset —
järjestelmät tulevat vaipumaan unhotuksen mereen, tulevat tyhjäksi
tehdyiksi, jota tuo "Maanalainen huone" eli "Kuilu" myöskin
vertauskuvaa. Sillä me pidämme "Kuilua" vertauskuvana ei
ainoastaan siitä suuresta hädästä, joka tulee kukistamaan ja
tekemään tyhjäksi nykyisen asiain tilan (seurauksena siitä, ettei se
ole sopusoinnussa sen paremman asiain järjestyksen kanssa, joka
saatetaan voimaan Jumalan valtakunnassa), vaan me pidämme sitä
myöskin sen varman lopun vertauskuvana, joka kohtaa jokaista
olentoa, joka jatkaa juoksemista tuolla alaspäin vievällä tiellä, ja joka
tuhatvuotiskauden täydestä valosta huolimatta kieltäytyy luopumasta
synneistään ja tavottelemasta vanhurskautta.
Huomaa tässä yhteydessä myöskin toinen asia:
"Sisäänkäyntikäytävä" kallistuu tasaisesti alaspäin, kunnes se
lähestyy "Kuilua", jolloin se lakkaa kallistumasta ja kulkee

vaakasuorasti. Onnettomuudeksi emme löydä mitään tarkkaa mittaa
tälle käytävän osalle. Vahvistamaton mitta on 324 tuumaa; joka
antaisi vuoden 1590 tai Shakespearen ajan. Kuitenkaan emme
kiinnitä mitään huomiota siihen. Yksi asia on varma — tämä matala
alaspäin menevä käytävä esittää maailman kulkua, kuten ylöspäin
johtava käytävä "Suuressa galleriassa" osotti "kutsutun"
seurakunnan kulkua. Muutos alaspäin johtavasta vaakasuoraan
suuntaan näyttää sentähden merkitsevän siveellistä tai valtiollista
valistusta, tai suosiollista alaspäin menevän matkan hillitsemistä.
Kuudennentoista vuosisadan uskonpuhdistus sai välillisesti
varmaan paljon aikaan nostaakseen maailmaa joka suhteessa. Se
puhdisti siveellisen ilmapiirin sen paljosta tietämättömyydestä ja
taikauskosta, ja roomalais-katolilaiset samoinkuin protestantitkin
myöntävät sen merkitsevän uutta aikaa yleisessä edistyksessä.
Me emme väitä, kuten muutamat, että kaikki meidän päivinämme
johtaisi pikemmin ylöspäin kuin alaspäin. Päinvastoin, me näemme
monta asiaa meidän päivinämme, joihin ei voi sivistyksenkään
kannalta suostua, puhumattakaan niiden sopusoinnusta Jumalan
tahdon kanssa. Me näemme vapaamman "inhimillisen"
katsantokannan olevan vallalla maailmassa, joka, joskin kaukana
Herramme Jeesuksen uskonnosta, on pitkälle edistynyt verrattuna
menneisyyden tietämättömään taikauskoon.
Todellakin on tämä yhteiskunnallinen parannus maailmassa
antanut aihetta "kehitysopille" ja saattanut monet päättämään, että
maailma on nopeasti tulemassa paremmaksi ja paremmaksi, — ettei
se tarvitse mitään pelastajaa ja hänen lunastavaa työtänsä, eikä
myöskään valtakuntaa ennalleenasettamis-siunauksineen. Hyvin pian
tulee maailma-parka huomaamaan, että kohoaminen ja puhtaan

itsekkäisyyden perustus merkitsevät kasvavaa tyytymättömyyttä, ja
mahdollista laittomuutta. Ainoastaan Herran kansa, hänen sanansa
valaisemana, kykenee näkemään asiat niiden oikeassa valossa.
Mutta ylläkosketeltujen mittain antaessa yhtäpitävät todistukset,
oli eräs mitta, joka ei näyttänyt sopivan yhteen Raamatun esityksen
kanssa, nim. "Ensimäisen nousevan käytävän" mitta, joka nähtävästi
esitti aikaa Israelin Egyptistä lähdöstä Herramme Jeesuksen
syntymään saakka. [Tämä ajanjakso ei ole sama kuin se, minkä
olemme maininneet II osassa. 7 luvussa ja esittäneet juutalaiseksi
aikakaudeksi. Tämä jälkimäinen alkoi 198 vuotta ennen Egyptistä
lähtöä Jaakobin kuollessa eikä loppunut ennenkuin Herramme, jonka
he hylkäsivät, jätti heidän huoneensa heille autioiksi viisi päivää
ennen ristiinnaulitsemistaan.] Raamatun kertomusta ajasta sellaisena
kuin se on jo ilmotettu [II osa, siv. 269—278], emme voineet epäillä,
koska me olimme osottaneet sen oikeaksi niin monella tavalla. Se
osotti että aika Egyptistä lähdöstä vuoteen 1 oli tasan 1614 vuotta,
mutta "Ensimäisen nousevan käytävän" lattialinja on ainoastaan
1,542 tuumaa. Edelleen tiesimme Herramme ja profettain sanoista
varmasti, että lakikausi ja lihallisen Israelin suosio ei loppunut
Jeesuksen syntyessä, vaan kolme ja puoli vuotta hänen kuolemansa
jälkeen, heidän 70 suosion viikkonsa lopussa, vuonna 36 j.K. [II osa,
7 luku] Täten tekee aikajakso Egyptistä lähdöstä heidän suosionsa
täydelliseen loppuun (1614 + 36) =1650 vuotta. Ja vaikka eräässä
tarkotuksessa tuo suuremmoinen ja siunauksesta rikas uusi
armotalous alkoikin Jeesuksen syntymässä (Luukk. 2: 10—14; 25—
38) pitäisi suuren pyramiidin kumminkin jollakin tavoin ilmaista
Israelin suosion täysi pituus. Tämän huomasimmekin me ilmaistuksi
mitä älykkäämmällä tavalla; graniitti "Tulppa" osottautui olevan
kylliksi pitkä täyttämään tämä aikakausi juuri loppuunsa. Silloin
tiesimme me minkätähden tämä "Tulppa" oli pantu niin lujasti kiinni

ettei kenenkään ollut onnistunut irrottaa sitä. Suuri Rakennusmestari
oli asettanut sen sinne, niin ettei se murtuisi, jotta me nyt saisimme
kuulla sen todistuksen vahvistavan Raamatun sekä sen
suunnitelmaan että aikalaskuihin nähden.
Mitatessamme tätä käytävää ja "Tulppaa" tulee meidän ajatella
sitä kiikariksi tai kaukoputkeksi, josta "tulppa" on vedetty ulos niin
pitkälle että sen ylin pää on tullut siihen paikkaan, missä alunperin
sen alin pää on ollut. Etäisyys "suuren gallerian" pohjoisesta
sisäänkäytävästä alaspäin graniitti-"tulpan" alimpaan päähän asti on
1,470 tuumaa ja jos me tähän lisäämme "tulpan" pituuden 179
tuumaa, saamme kokonaissummaksi 1,649 tuumaa, joka vastaa
1649 vuotta; ja yhden tuuman erotus tästä saadun luvun ja niiden
1650 vuoden välillä, jotka Raamatullinen aikalasku ilmottaa tämän
aikakauden pituudeksi, on helposti selitettävissä, kun me muistamme
että tuon graniitti-"tulpan" toista päätä ovat melkoisesti hanganneet
ne, jotka koettivat saada sen pois kiinteältä paikaltaan käytävässä.
Niin ollen vahvistaa tämä kivi-"todistaja" täydellisesti Raamatun
todistuksen, ja osottaa että aikajakso Israelin Egyptistä lähdöstä
heidän kansallisen suosionsa täydelliseen loppuun asti [II osa, 3
luku], vuonna 36 j.K. oli 1650 vuotta. Mutta älköön kukaan sekottako
tätä aikajaksoa sen aikajakson kanssa, joka esitetään juutalaisen ja
kristillisen armotalouden rinnakkaisuuksissa edellisessä osassa —
missä osotetaan että nämä aikakaudet ovat kumpikin 1845 vuotta
pitkät, toinen Jaakobin kuolemasta vuoteen 33 j.K. ja toinen
vuodesta 33 j.K. vuoteen 1878.
Eikä siinä kyllin että tämä oli älykäs tapa kätkeä ja kumminkin
ilmaista aika Egyptistä lähdöstä Vapahtajamme syntymään (jotta se
määrättynä aikana vahvistaisi Raamatun todistuksen), vaan

tarkkaavaisen lukijan pitäisi ilman vaikeutta huomata, että se
kahdestakin syystä voi tapahtua ainoastaan jollakin samallaisella
tavalla. Ensiksikin, koska juutalainen armotalous ja suosio ei
ainoastaan alkanut Jaakobin kuollessa, ennen Egyptistä lähtöä, vaan
myöskin ulottui kristilliseen armotalouteen ja kulki sen kanssa
rinnatusten niinä kolmenakymmenenä kolmena vuotena, jolloin
Herramme eli maan päällä; ja toiseksi, jos "Ensimäinen nouseva
käytävä" olisi tehty kyllin pitkäksi, jotta se olisi täydellisesti esittänyt
juutalaisen ajan vuosi-tuumissa, olisi käynyt välttämättömäksi tehdä
pyramiidi vielä suuremmaksi, mikä taas olisi hävittänyt sen
tieteelliset piirteet ja opetukset.
Tutkikaamme nyt "Suurta galleriaa" "Ensimäisen nousevan
käytävän" loppukohdasta ja pankaamme merkille myöskin sen
vertauskuvalliset todistukset. Se on seitsemän kertaa korkeampi kuin
"Ensimäinen nouseva käytävä". Sen seinissä on seitsemän kerrosta
toistensa yli työntyviä kiviä, tasaisesta, hienoksi kiillotetusta kivestä
ja kerran kauniista, kermanvärisestä kalkkikivestä. Se on
kaksikymmentä kahdeksan jalkaa (n. 8,4 m) korkea, vaikkakin hyvin
kapea, senkautta ettei se ole ainoassakaan kohdassa kuutta jalkaa
(n. 1,8 m) leveämpi, mutta supistuen alaspäin on se lattian kohdalta
ainoastaan kolme jalkaa (n. 0,9 m) ja katosta vieläkin soukempi.
Prof. Greaves, professori Oxfordissa viidennellätoista vuosisadalla,
kirjotti selonteossaan siitä: —
"Tämä on hyvin komea työ eikä ole huonompi taiteelliseen
huomattavaisuuteen tai aineelliseen rikkauteen nähden mitä
komeampia ja suuremmoisempia rakennuksia. Tämä galleria eli
korridoori tai miksi sitä kutsuisin on rakennettu valkosesta,
kiillotetusta marmorista (kalkkikivestä), joka on leikattu hyvin
tasaisiksi nelikulmaisiksi laatoiksi. Samallaisesta aineesta kuin lattia,

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Infogenomics The Informational Analysis Of Genomes Vincenzo Manca

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